{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0001","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6659fe57f5473861ad488679cd478c7149c5e554cea0eb923a789965c29f907a","family_id":"qf-mod34-modinv-existence","seed":202611037001,"semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":27}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=2 and modulus n=27≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(2, 27, 1, Yes, 14)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (2, 27, 1, Yes, 14)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0002","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32d2ea481885d7c463220e2f061f4777e22965d6eea8481b6c3953f9be1b928a","family_id":"qf-mod34-modinv-existence","seed":202611037002,"semantic_template":"qf-mod34-modinv-existence/{\"a\":3,\"n\":28}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=3 and modulus n=28≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(3, 28, 1, Yes, 19)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (3, 28, 1, Yes, 19)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0003","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"481940c13b5952d1a2491712d4e11e64ecb6863dacb62d03f96fa00ca99755da","family_id":"qf-mod34-modinv-existence","seed":202611037003,"semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":29}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=2 and modulus n=29≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(2, 29, 1, Yes, 15)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (2, 29, 1, Yes, 15)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0004","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4eeae290ce7bb623b7d669628a9724aa3956d0d524c6282716c39b79be72e4eb","family_id":"qf-mod34-modinv-existence","seed":202611037004,"semantic_template":"qf-mod34-modinv-existence/{\"a\":7,\"n\":30}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=7 and modulus n=30≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(7, 30, 1, Yes, 13)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (7, 30, 1, Yes, 13)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0005","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"026a6e4a3b983b78892ba028b1d7c778b507cf9d6224f52d8074e902e3de75c9","family_id":"qf-mod34-modinv-existence","seed":202611037005,"semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":31}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=2 and modulus n=31≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(2, 31, 1, Yes, 16)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (2, 31, 1, Yes, 16)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0006","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61afec91bbb8f32957d1e7ffb3552ec2aae58a64b3a2d551496c21461b92acb4","family_id":"qf-mod34-modinv-existence","seed":202611037006,"semantic_template":"qf-mod34-modinv-existence/{\"a\":3,\"n\":32}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=3 and modulus n=32≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(3, 32, 1, Yes, 11)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (3, 32, 1, Yes, 11)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0007","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9393f2a48013ce26b1fffee622653f973a42eece47df4cd6bb310b99f5dd633","family_id":"qf-mod34-modinv-existence","seed":202611037007,"semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":33}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=2 and modulus n=33≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(2, 33, 1, Yes, 17)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (2, 33, 1, Yes, 17)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0008","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"451ef901562faaab52fa5dc18c439ea4b34a56022bcdb3210a234749ea1e3af4","family_id":"qf-mod34-modinv-existence","seed":202611037008,"semantic_template":"qf-mod34-modinv-existence/{\"a\":3,\"n\":34}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=3 and modulus n=34≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(3, 34, 1, Yes, 23)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (3, 34, 1, Yes, 23)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0009","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d2e86f117297eb93cc7be82ab81c7aca407bb812f195fe690661066ff4ca71f","family_id":"qf-mod34-modinv-existence","seed":202611037009,"semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":35}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=2 and modulus n=35≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(2, 35, 1, Yes, 18)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (2, 35, 1, Yes, 18)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0010","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9db0e87fa7298a4cc001984d240ec0e1af1a91a4157c3a4b2ee08cc52d51c8d","family_id":"qf-mod34-modinv-existence","seed":202611037010,"semantic_template":"qf-mod34-modinv-existence/{\"a\":5,\"n\":36}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=5 and modulus n=36≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(5, 36, 1, Yes, 29)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (5, 36, 1, Yes, 29)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0011","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"308abe01d2d62998bf26feccf5a465cbd7f80bb56950d8b5b0d48a989340877b","family_id":"qf-mod34-modinv-existence","seed":202611037011,"semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":37}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=2 and modulus n=37≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(2, 37, 1, Yes, 19)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (2, 37, 1, Yes, 19)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0012","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9331b78b71b9f3c1f65e8cf451d0d4a7fb30869e82a9c6dd074e225643f3a0f","family_id":"qf-mod34-modinv-existence","seed":202611037012,"semantic_template":"qf-mod34-modinv-existence/{\"a\":3,\"n\":38}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=3 and modulus n=38≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(3, 38, 1, Yes, 13)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (3, 38, 1, Yes, 13)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0013","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"060c74577ce70fc20470c177268d1012acf26a8aaa46bfc5d07ce5f1a4e0b5bd","family_id":"qf-mod34-modinv-existence","seed":202611037013,"semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":39}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=2 and modulus n=39≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(2, 39, 1, Yes, 20)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (2, 39, 1, Yes, 20)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0014","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05857d6ed89954a69fff02fec77931b33a5eade41b602c794225f239aec75797","family_id":"qf-mod34-modinv-existence","seed":202611037014,"semantic_template":"qf-mod34-modinv-existence/{\"a\":3,\"n\":40}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=3 and modulus n=40≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(3, 40, 1, Yes, 27)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (3, 40, 1, Yes, 27)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0015","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee1542e5590f573ab1e0dff72042d3001abf386b483c8b88d4c1015c587e27f5","family_id":"qf-mod34-modinv-existence","seed":202611037015,"semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":41}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=2 and modulus n=41≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(2, 41, 1, Yes, 21)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (2, 41, 1, Yes, 21)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0016","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27c345cb7a3fb334deb3761712b6657f1e22bbc9ce6cf62f2393f7bd6f5054aa","family_id":"qf-mod34-modinv-existence","seed":202611037016,"semantic_template":"qf-mod34-modinv-existence/{\"a\":5,\"n\":42}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=5 and modulus n=42≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(5, 42, 1, Yes, 17)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (5, 42, 1, Yes, 17)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0017","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef545aa9391e194bda9d0f855c5bd892a308739cba362463adc484edeff80aa7","family_id":"qf-mod34-modinv-existence","seed":202611037017,"semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":43}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=2 and modulus n=43≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(2, 43, 1, Yes, 22)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (2, 43, 1, Yes, 22)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0018","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"848fe4a352f2bc0f7f6a88cd5d65629bbccfeae275f6f308f189f12626d13b0d","family_id":"qf-mod34-modinv-existence","seed":202611037018,"semantic_template":"qf-mod34-modinv-existence/{\"a\":3,\"n\":44}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=3 and modulus n=44≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(3, 44, 1, Yes, 15)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (3, 44, 1, Yes, 15)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0019","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f099bd6397b39aa03c392179f2e57449d6419a06d278c6a238956d3fc498952b","family_id":"qf-mod34-modinv-existence","seed":202611037019,"semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":45}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=2 and modulus n=45≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(2, 45, 1, Yes, 23)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (2, 45, 1, Yes, 23)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0020","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ec29258488ba583005f8b049190b06e2b15ebd0031e721d1385df9c405c0d71","family_id":"qf-mod34-modinv-existence","seed":202611037020,"semantic_template":"qf-mod34-modinv-existence/{\"a\":3,\"n\":46}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=3 and modulus n=46≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(3, 46, 1, Yes, 31)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (3, 46, 1, Yes, 31)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0021","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76eca3bd7fe020e646d3a4550b4c632094d51ceb9205dc6eb49a3a81eaa6d1cc","family_id":"qf-mod34-modinv-existence","seed":202611037021,"semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":47}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=2 and modulus n=47≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(2, 47, 1, Yes, 24)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (2, 47, 1, Yes, 24)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0022","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7615350c533b4d6e24c4ffc21d74d561b93ffaa048052f5ba23731508c9d5217","family_id":"qf-mod34-modinv-existence","seed":202611037022,"semantic_template":"qf-mod34-modinv-existence/{\"a\":5,\"n\":48}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=5 and modulus n=48≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(5, 48, 1, Yes, 29)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (5, 48, 1, Yes, 29)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0023","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5378f2c47790732c65e28bcdb20e6636875b3574569fda6b8ff44a06cc4e32cb","family_id":"qf-mod34-modinv-existence","seed":202611037023,"semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":49}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=2 and modulus n=49≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(2, 49, 1, Yes, 25)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (2, 49, 1, Yes, 25)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0024","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adc8a968ac9ded8fe48645a9865bb40dd15486903d4e724f39900318dbc61a1a","family_id":"qf-mod34-modinv-existence","seed":202611037024,"semantic_template":"qf-mod34-modinv-existence/{\"a\":3,\"n\":50}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=3 and modulus n=50≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(3, 50, 1, Yes, 17)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (3, 50, 1, Yes, 17)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0025","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86f919053aea4d72608cf67dea661b3c0751f9f759e977c348b545285383d9df","family_id":"qf-mod34-modinv-existence","seed":202611037025,"semantic_template":"qf-mod34-modinv-existence/{\"a\":2,\"n\":51}"},"fields":{"title":"A Modular Inverse Exists Exactly for a Unit","stem":"Use the original signed integer a=2 and modulus n=51≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,exists?,inverse), with inverse=null for a nonunit.","answer":{"kind":"expression","value":"(2, 51, 1, Yes, 26)","explanation":"Compute the exact gcd first. A unit obtains its normalized inverse from the Bezout coefficient; a gcd greater than one excludes an inverse modulo n. The exact result, retaining the requested order, is (2, 51, 1, Yes, 26)."},"skills":["A Modular Inverse Exists Exactly for a Unit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0026","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a644bb668d88eda1a6db8c469a65442e0ec461ab9ad04fe626b82cc0ee3abd73","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037026,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":27}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=2 and modulus n=27≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(2, 27, 1, -13, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (2, 27, 1, -13, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0027","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d5a4419846c76298af8030c4611bfd15adb584837423a5c5849d94fe4a67191","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037027,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":3,\"n\":28}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=3 and modulus n=28≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(3, 28, 1, -9, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (3, 28, 1, -9, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0028","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a499f11dff12bb327ce2cc0bcdb2f633c0589244eb2e2daf8d81732261903122","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037028,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":29}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=2 and modulus n=29≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(2, 29, 1, -14, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (2, 29, 1, -14, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0029","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d72c6e0e2026d85f9ea18e9c4bce8c3c41e55b9b6bdc681adef65260ca19cdb","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037029,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":7,\"n\":30}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=7 and modulus n=30≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(7, 30, 1, 13, -3, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (7, 30, 1, 13, -3, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0030","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"161c3c4bb3cfd46372c164d82c7560dc09649bd65bfe20d2a6a8d6533ac7f5a7","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037030,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":31}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=2 and modulus n=31≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(2, 31, 1, -15, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (2, 31, 1, -15, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0031","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32d46fa0e6f63d096c5d47c058379ffd22d30b84844d2fc7a25959a9de435d34","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037031,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":3,\"n\":32}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=3 and modulus n=32≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(3, 32, 1, 11, -1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (3, 32, 1, 11, -1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0032","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c43ff812a7a95c996c34a33866d5d27919f305a7da4a7e84032896acffa4138","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037032,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":33}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=2 and modulus n=33≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(2, 33, 1, -16, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (2, 33, 1, -16, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0033","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55f33767b0e5277e015f5d7bb649b368005692e3c1bf963b1538b00e746c05db","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037033,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":3,\"n\":34}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=3 and modulus n=34≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(3, 34, 1, -11, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (3, 34, 1, -11, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0034","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"936cb4ce1f64db51412f748c69eff5d841d68acf446ea2a6633b14432cfe8375","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037034,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":35}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=2 and modulus n=35≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(2, 35, 1, -17, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (2, 35, 1, -17, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0035","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ad58935f58b8b9b52900766a517b21391c5edc2a007983e21ed18280f3ca7ce","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037035,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":5,\"n\":36}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=5 and modulus n=36≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(5, 36, 1, -7, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (5, 36, 1, -7, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0036","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad5add017a0aad9a3f7420bb347def1dca7c03ecbd93a9f08d307a97a031969b","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037036,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":37}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=2 and modulus n=37≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(2, 37, 1, -18, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (2, 37, 1, -18, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0037","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b29700810b41d9b317ef22667e057ab6a17b3c351537c4d8ae6e928a627218cc","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037037,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":3,\"n\":38}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=3 and modulus n=38≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(3, 38, 1, 13, -1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (3, 38, 1, 13, -1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0038","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f4227125383c591cbfbd7ee769185a7c286e2fa9ecc19a134e58412cdf2b88c","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037038,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":39}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=2 and modulus n=39≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(2, 39, 1, -19, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (2, 39, 1, -19, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0039","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0db6d347aa1948d4ed56e1ab7b2466074327c958b73fe30205c36a7e27acf6b","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037039,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":3,\"n\":40}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=3 and modulus n=40≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(3, 40, 1, -13, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (3, 40, 1, -13, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0040","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80cfefdad9f4228399a6ce8732c298d65ae5a200278827eb87a1f1898cc6b1db","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037040,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":41}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=2 and modulus n=41≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(2, 41, 1, -20, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (2, 41, 1, -20, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0041","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea4cd6edd096b37f81790ec8a1efe09754411ea657d4e230080f12b0a9fc4010","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037041,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":5,\"n\":42}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=5 and modulus n=42≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(5, 42, 1, 17, -2, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (5, 42, 1, 17, -2, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0042","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"860660296afd8efc494f7a23c7b6104618203e6cdb613e6b1fcfe3f0e8945243","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037042,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":43}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=2 and modulus n=43≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(2, 43, 1, -21, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (2, 43, 1, -21, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0043","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1643d373920bd59e3fce01ababac021efed7c94c866575ed1c44d9aa1c12557","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037043,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":3,\"n\":44}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=3 and modulus n=44≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(3, 44, 1, 15, -1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (3, 44, 1, 15, -1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0044","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03926c5d4af308769de588491da5cd8fa2517e5a2481f18d8570d71cee3681ba","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037044,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":45}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=2 and modulus n=45≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(2, 45, 1, -22, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (2, 45, 1, -22, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0045","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"178c6138aed6c8b11ac01f057a1dcf57ed4d7153638a33d9fe97707d5cf7a4e9","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037045,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":3,\"n\":46}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=3 and modulus n=46≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(3, 46, 1, -15, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (3, 46, 1, -15, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0046","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d9e81ea15cdf70e28e5eb899a12bcd60d107788ed83eb9c5174c305054ea42c","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037046,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":47}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=2 and modulus n=47≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(2, 47, 1, -23, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (2, 47, 1, -23, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0047","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"652e15acd0b2838b561efc3794dfed7d6a92466ba078849d88f1b7e59770535a","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037047,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":5,\"n\":48}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=5 and modulus n=48≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(5, 48, 1, -19, 2, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (5, 48, 1, -19, 2, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0048","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59bff0a6ad05ade6797903a483615023293557c569d811eca88e627976229c66","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037048,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":49}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=2 and modulus n=49≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(2, 49, 1, -24, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (2, 49, 1, -24, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0049","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96f67ddb30c063eea29a173f6ac036af4b697540a2b1a5787a95b9c968518895","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037049,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":3,\"n\":50}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=3 and modulus n=50≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(3, 50, 1, 17, -1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (3, 50, 1, 17, -1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0050","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b97352b7c70e90feb4f81e76c60ea540e06b973686a9cc39dabe1b108fce82a","family_id":"qf-mod34-modinv-extended-euclid","seed":202611037050,"semantic_template":"qf-mod34-modinv-extended-euclid/{\"a\":2,\"n\":51}"},"fields":{"title":"Keep Signed Bezout Coefficients and Verify the Integer Identity","stem":"Use the original signed integer a=2 and modulus n=51≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Give (a,n,gcd,x,y,bezout_value,classification), retaining the coefficients returned by exact Euclidean steps on |a| and n, with the sign of a restored in x.","answer":{"kind":"expression","value":"(2, 51, 1, -25, 1, 1, Bezout_equals_one)","explanation":"Substitute the signed coefficients into a*x+n*y. The value is the nonnegative gcd. Classification is Bezout_equals_one or nonunit_Bezout_gcd; the coefficients are witnesses, not a unique pair of integers. The exact result, retaining the requested order, is (2, 51, 1, -25, 1, 1, Bezout_equals_one)."},"skills":["Keep Signed Bezout Coefficients and Verify the Integer Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0051","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6420a4f5105c6d8509d1940678d52996eb4016b94d647897bf09ba0969f8cdc9","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037051,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":27}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=2 and modulus n=27≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(2, 27, 14, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (2, 27, 14, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0052","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44c6325a035db91e540d27ccecebd239796e8113a7f8b644c727e7ba27d233fb","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037052,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":3,\"n\":28}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=3 and modulus n=28≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(3, 28, 19, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (3, 28, 19, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0053","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df24f14e6c760cb58ba7313fa0bff7713fb0d381c3107427fa1258142001f762","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037053,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":29}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=2 and modulus n=29≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(2, 29, 15, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (2, 29, 15, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0054","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc016b630d9f6444aa1cbaaa2bb9ad266c9dbc0cc38cb5127219ada080150405","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037054,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":7,\"n\":30}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=7 and modulus n=30≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(7, 30, 13, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (7, 30, 13, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0055","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee1eea087f8657f6bfd15f7547583e13c3dba8a4cc6ac6342f6d3f4a88473898","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037055,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":31}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=2 and modulus n=31≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(2, 31, 16, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (2, 31, 16, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0056","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66921985488b7fa0c1434dc48652bdfcef8fbd3c75a775cc603538bdda16499a","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037056,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":3,\"n\":32}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=3 and modulus n=32≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(3, 32, 11, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (3, 32, 11, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0057","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b7acc658dc30b156470d249bef515304387def460e3caab327457e807720d52","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037057,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":33}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=2 and modulus n=33≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(2, 33, 17, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (2, 33, 17, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0058","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee4c646ed1e9251e77c3c236e31eca4d422b2246c050a315e2754dfc5ea61e62","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037058,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":3,\"n\":34}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=3 and modulus n=34≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(3, 34, 23, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (3, 34, 23, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0059","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7107a4e9ee99dfed1d8317e52aa53e5b774c92652f22705067be24e0b0660788","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037059,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":35}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=2 and modulus n=35≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(2, 35, 18, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (2, 35, 18, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0060","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b4f6f23f0cad1c1b47df17d80bfb203c3af890f5d950e784a417998e2fa5a6e","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037060,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":5,\"n\":36}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=5 and modulus n=36≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(5, 36, 29, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (5, 36, 29, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0061","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3040b94d9126ed2c987a3515d972fb32f42f17c5b4e8d1cc9ef800c235c9fdff","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037061,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":37}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=2 and modulus n=37≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(2, 37, 19, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (2, 37, 19, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0062","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abb4b298b1487e114aef3c3f18b40062582c64012f9dc8837c44382d12b6378e","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037062,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":3,\"n\":38}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=3 and modulus n=38≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(3, 38, 13, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (3, 38, 13, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0063","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2034c7fa9674a1d025418ca487014d6685ce5959d66f61542fcf2af0f033cf07","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037063,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":39}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=2 and modulus n=39≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(2, 39, 20, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (2, 39, 20, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0064","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f8a95e7af4baf6941cfb49796627aede6fe1f091f6c1b7990abeb0573c2229e","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037064,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":3,\"n\":40}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=3 and modulus n=40≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(3, 40, 27, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (3, 40, 27, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0065","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07dce708345b601092ba879c2cb6105516411dabc9ba7fd4cb94228e48dbedb9","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037065,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":41}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=2 and modulus n=41≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(2, 41, 21, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (2, 41, 21, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0066","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdd1b1029cd70b6339ee0bd88e402854931bbfcdfcff013c5c8d9abb54cfe6b1","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037066,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":5,\"n\":42}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=5 and modulus n=42≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(5, 42, 17, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (5, 42, 17, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0067","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84a3648fcaf3a51f8ad0e33bda80dfdbd77a2274f0ebf2b1cc8b35f2e9638440","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037067,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":43}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=2 and modulus n=43≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(2, 43, 22, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (2, 43, 22, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0068","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac70a734e0bf1e6920b9e9082a11a00bd1ff5452b56da6aa6d16b5dd8b633c2f","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037068,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":3,\"n\":44}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=3 and modulus n=44≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(3, 44, 15, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (3, 44, 15, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0069","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"560635a2c92330ab16ddb52e8402626013976291a60361ad42e5c811efc87bb2","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037069,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":45}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=2 and modulus n=45≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(2, 45, 23, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (2, 45, 23, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0070","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcf8434840b8041342455cd8c841199fc1a70000dc48437d4ebbff5b1707d92e","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037070,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":3,\"n\":46}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=3 and modulus n=46≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(3, 46, 31, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (3, 46, 31, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0071","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97ee63b799d59f076f3002cb01651fdbf9e054ec87c09e1ea4dd32c4a3b107eb","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037071,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":47}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=2 and modulus n=47≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(2, 47, 24, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (2, 47, 24, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0072","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1014ac7996922e2ca5e6852cbe32a57f6e94038595bc5ae9968172049645e57d","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037072,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":5,\"n\":48}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=5 and modulus n=48≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(5, 48, 29, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (5, 48, 29, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0073","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"871d7cf088c22f0acfa26f4e07dc8fc50906dcd34d55492f9e9ee5d581ed62b4","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037073,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":49}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=2 and modulus n=49≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(2, 49, 25, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (2, 49, 25, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0074","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8da5cd290b3a4e187232ac0370a8f8a4eff4df67c4e804cc7309ae105ef4bb43","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037074,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":3,\"n\":50}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=3 and modulus n=50≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(3, 50, 17, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (3, 50, 17, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0075","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a5d373cd0829654f0593178bd3b7e0229f0b032bd127afe5e92c79afa168c34","family_id":"qf-mod34-modinv-normalized-inverse","seed":202611037075,"semantic_template":"qf-mod34-modinv-normalized-inverse/{\"a\":2,\"n\":51}"},"fields":{"title":"Normalize the Inverse Representative, Not the Bezout Coefficients","stem":"Use the original signed integer a=2 and modulus n=51≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)=1. Give (a,n,inverse,product_mod_n,valid?,unique_modulo_n).","answer":{"kind":"expression","value":"(2, 51, 26, 1, Yes, unique_modulo_n)","explanation":"Reduce the exact Bezout x modulo n and check a*inverse≡1 mod n. The normalized representative is unique in [0,n), while infinitely many congruent integer representatives exist. The exact result, retaining the requested order, is (2, 51, 26, 1, Yes, unique_modulo_n)."},"skills":["Normalize the Inverse Representative, Not the Bezout Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0076","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c979bb6df141fd1574aa5591d94ad1eae57a754ff211bda894b819b60db90368","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037076,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":2,\"n\":4}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=2 and modulus n=4≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(2, 4, 2, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (2, 4, 2, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0077","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6de0750407f86691c223f13158ff35ba06b14b70afd2e2707f68e3dd0b487c6","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037077,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":3,\"n\":6}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=3 and modulus n=6≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(3, 6, 3, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (3, 6, 3, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0078","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4b292349dfeae85b68cb1de3fda2bc68d75bdcf3f212e3663944e32f5df8e32","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037078,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":4,\"n\":8}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=4 and modulus n=8≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(4, 8, 4, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (4, 8, 4, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0079","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fee845931a20bcd24ae5cc92c618ad44e13369c91c43e88b108a9ebc7e93fb9","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037079,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":5,\"n\":10}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=5 and modulus n=10≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(5, 10, 5, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (5, 10, 5, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0080","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53ff4ccca053f34fd37cc492cd6044e6f4095985464aa38995c4918873b8a56b","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037080,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":6,\"n\":12}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=6 and modulus n=12≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(6, 12, 6, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (6, 12, 6, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0081","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df623f67aba8d7cc869aa4f75663e17696ed89d04c4353f6e99fe7993f2297c9","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037081,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":7,\"n\":14}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=7 and modulus n=14≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(7, 14, 7, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (7, 14, 7, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0082","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0ee499d126d1a661ddead06ee0ca25d1f071a27c2d61997016e18c0a70c181b","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037082,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":8,\"n\":16}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=8 and modulus n=16≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(8, 16, 8, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (8, 16, 8, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0083","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"479a52ac87ac570364686486c5b1bd0ab95951b7d983c8e179410f20ce3583b1","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037083,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":9,\"n\":18}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=9 and modulus n=18≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(9, 18, 9, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (9, 18, 9, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0084","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3eff7d311ac921747a72e5248a0de6fac290a886831e57f16dd41deee7a76c28","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037084,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":10,\"n\":20}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=10 and modulus n=20≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(10, 20, 10, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (10, 20, 10, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0085","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52a5fca10ea5134decdbdfe806dc11a585cfc16d738e537bb70f7ed74031e732","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037085,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":11,\"n\":22}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=11 and modulus n=22≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(11, 22, 11, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (11, 22, 11, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0086","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58b61142a362f81024104aecdcbf42cc66d45bc6b6cfad5ad14ce753fa427347","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037086,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":12,\"n\":24}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=12 and modulus n=24≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(12, 24, 12, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (12, 24, 12, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0087","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b674830744129304f4f3068f8a7024d22a2745878512c8a055cb6ab13f2cdeea","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037087,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":13,\"n\":26}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=13 and modulus n=26≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(13, 26, 13, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (13, 26, 13, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0088","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2e7aef8be06cb556dd40a568b587f9428476584fd983b0418be2783049e8679","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037088,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":14,\"n\":28}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=14 and modulus n=28≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(14, 28, 14, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (14, 28, 14, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0089","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f48200e027c8b60f95be911aa4614d1819afda837d0e732c67f8f4e4e6ee7de2","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037089,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":15,\"n\":30}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=15 and modulus n=30≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(15, 30, 15, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (15, 30, 15, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0090","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c38847342f9dbc19e0a018e8a2df0c370ebb8814f829b68014ec115ec98140d","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037090,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":16,\"n\":32}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=16 and modulus n=32≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(16, 32, 16, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (16, 32, 16, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0091","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"114c371dd2a9dd306d360ebf9999cfe7fb98787e841e02aef0788d328cde8dba","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037091,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":17,\"n\":34}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=17 and modulus n=34≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(17, 34, 17, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (17, 34, 17, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0092","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8de927847351d6c5e2fe86899d292d143b19d4be756689b66c7c807e1c0d2a3a","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037092,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":18,\"n\":36}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=18 and modulus n=36≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(18, 36, 18, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (18, 36, 18, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0093","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc2e9d07a1749ce1a3a9cb749ee87837a1e5843d37484bfdfd394c8d3ce61687","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037093,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":19,\"n\":38}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=19 and modulus n=38≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(19, 38, 19, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (19, 38, 19, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0094","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49d6f074f47f5ea6e5437a1e44c8991ab3e52ba1f3b8cc29d8a290441435a089","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037094,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":20,\"n\":40}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=20 and modulus n=40≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(20, 40, 20, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (20, 40, 20, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0095","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e137e511ad859297543588e2a7fc74e60df3e3e0ccfc5bf1a4bc84478a9b6165","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037095,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":21,\"n\":42}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=21 and modulus n=42≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(21, 42, 21, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (21, 42, 21, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0096","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1e41c98032d5a1d0b28f0754772a474f9adf0caff92364eefab4d1076e26c72","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037096,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":22,\"n\":44}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=22 and modulus n=44≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(22, 44, 22, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (22, 44, 22, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0097","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f27186aafda35cd8b3e067fd10d81b885a1dbfa7ebffe35ab3a2958902ad1944","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037097,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":23,\"n\":46}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=23 and modulus n=46≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(23, 46, 23, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (23, 46, 23, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0098","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f07b329f4decb61b8c782a4e28be1a83e695606a7edfc9ccbd55c7d273c87a26","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037098,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":24,\"n\":48}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=24 and modulus n=48≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(24, 48, 24, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (24, 48, 24, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0099","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3ab0b56e39a6fc33817e0db96a3d3e86e427034713617036d92d382c3841957","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037099,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":25,\"n\":50}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=25 and modulus n=50≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(25, 50, 25, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (25, 50, 25, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r07-modular-inverse-0100","batch_id":"batch-20261014-advance-20261103-r07-modular-inverse-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d0eba1c739a2f4f72f5c7ee1482c55d63bd8b6fc45d6d87cd71485acb0b5741","family_id":"qf-mod34-modinv-nonunit-boundary","seed":202611037100,"semantic_template":"qf-mod34-modinv-nonunit-boundary/{\"a\":26,\"n\":52}"},"fields":{"title":"A Nonunit Has No Modular Inverse","stem":"Use the original signed integer a=26 and modulus n=52≥2. Exact extended Euclid gives nonnegative g=gcd(a,n) and signed coefficients x,y satisfying a*x+n*y=g. An inverse exists exactly when g=1; normalize its representative to 0≤inverse<n. Uniqueness means one residue class modulo n, not a unique integer representative. This family excludes moduli 0 and 1 and does not use approximate division. Require gcd(a,n)>1. Give (a,n,gcd,inverse,exists?,nonunit_has_no_modular_inverse), with inverse=null.","answer":{"kind":"expression","value":"(26, 52, 26, null, No, nonunit_has_no_modular_inverse)","explanation":"A product congruent to one would force the gcd to divide one, contradicting this nonunit premise. Do not attempt ordinary or floating-point division, and reject a unit supplied to this nonunit-boundary family. The exact result, retaining the requested order, is (26, 52, 26, null, No, nonunit_has_no_modular_inverse)."},"skills":["A Nonunit Has No Modular Inverse"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}