# Question 948: The First Positive Power Returning to One Defines the Order [Return to catalogue page 5](https://ocme.evemisslab.com/en/questions/page/5/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations/#q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0048) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. P1000-0948 · q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0048 ## Original question and givens Use exact modulus n=25 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate). - G001 — Use exact modulus n=25 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate). ## Definition and scope Require n≥2 and a normalized residue coprime to n. The order is the first positive exponent returning to 1. - [mko-multiplicative-order-mod-n](https://ocme.evemisslab.com/en/mko/mko-multiplicative-order-mod-n/) ## Step-by-step derivation S001 · impliesBefore n=25; a=2 After order is defined here only for n≥2 and units modulo n Conditions: none Fix this node's domain first; do not confuse additive and multiplicative order. Step identifiers and dependenciesRule: order_domain · Uses: G001 S002 · impliesBefore a=2 After normalized residue=2; gcd(2,25)=1 Conditions: none Normalize even a negative representative into[0,n), then test coprimality. Step identifiers and dependenciesRule: normalize_and_qualify · Uses: G001, S001 S003 · impliesBefore gcd(residue,n)=1; units=[1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24] After phi(25)=20 Conditions: none Enumerate every coprime class from0 through n−1; phi is the complete unit-carrier size. Step identifiers and dependenciesRule: complete_unit_enumeration · Uses: G001, S002 S004 · calculationBefore power product at k=1 := mul(1, 2) After power product at k=1 = 2 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore product=2 After a^1 mod 25=2 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S004 S006 · calculationBefore power product at k=2 := mul(2, 2) After power product at k=2 = 4 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore product=4 After a^2 mod 25=4 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S006 S008 · calculationBefore power product at k=3 := mul(4, 2) After power product at k=3 = 8 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore product=8 After a^3 mod 25=8 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S008 S010 · calculationBefore power product at k=4 := mul(8, 2) After power product at k=4 = 16 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore product=16 After a^4 mod 25=16 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S010 S012 · calculationBefore power product at k=5 := mul(16, 2) After power product at k=5 = 32 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore product=32 After a^5 mod 25=7 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S012 S014 · calculationBefore power product at k=6 := mul(7, 2) After power product at k=6 = 14 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore product=14 After a^6 mod 25=14 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S014 S016 · calculationBefore power product at k=7 := mul(14, 2) After power product at k=7 = 28 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore product=28 After a^7 mod 25=3 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S016 S018 · calculationBefore power product at k=8 := mul(3, 2) After power product at k=8 = 6 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore product=6 After a^8 mod 25=6 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S018 S020 · calculationBefore power product at k=9 := mul(6, 2) After power product at k=9 = 12 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore product=12 After a^9 mod 25=12 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S020 S022 · calculationBefore power product at k=10 := mul(12, 2) After power product at k=10 = 24 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · impliesBefore product=24 After a^10 mod 25=24 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S022 S024 · calculationBefore power product at k=11 := mul(24, 2) After power product at k=11 = 48 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · impliesBefore product=48 After a^11 mod 25=23 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S024 S026 · calculationBefore power product at k=12 := mul(23, 2) After power product at k=12 = 46 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · impliesBefore product=46 After a^12 mod 25=21 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S026 S028 · calculationBefore power product at k=13 := mul(21, 2) After power product at k=13 = 42 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · impliesBefore product=42 After a^13 mod 25=17 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S028 S030 · calculationBefore power product at k=14 := mul(17, 2) After power product at k=14 = 34 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · impliesBefore product=34 After a^14 mod 25=9 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S030 S032 · calculationBefore power product at k=15 := mul(9, 2) After power product at k=15 = 18 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · impliesBefore product=18 After a^15 mod 25=18 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S032 S034 · calculationBefore power product at k=16 := mul(18, 2) After power product at k=16 = 36 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · impliesBefore product=36 After a^16 mod 25=11 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S034 S036 · calculationBefore power product at k=17 := mul(11, 2) After power product at k=17 = 22 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · impliesBefore product=22 After a^17 mod 25=22 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S036 S038 · calculationBefore power product at k=18 := mul(22, 2) After power product at k=18 = 44 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · impliesBefore product=44 After a^18 mod 25=19 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S038 S040 · calculationBefore power product at k=19 := mul(19, 2) After power product at k=19 = 38 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · impliesBefore product=38 After a^19 mod 25=13 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S040 S042 · calculationBefore power product at k=20 := mul(13, 2) After power product at k=20 = 26 Conditions: none Multiply the preceding power representative by the same normalized base. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · impliesBefore product=26 After a^20 mod 25=1 Conditions: none The remainder is the next full power representative. Step identifiers and dependenciesRule: reduce_next_power · Uses: G001, S042 S044 · impliesBefore all positive k<=20: [[1, 2], [2, 4], [3, 8], [4, 16], [5, 7], [6, 14], [7, 3], [8, 6], [9, 12], [10, 24], [11, 23], [12, 21], [13, 17], [14, 9], [15, 18], [16, 11], [17, 22], [18, 19], [19, 13], [20, 1]] After first positive return to1 occurs at k=20 Conditions: none Every smaller positive exponent has been checked not to return to1; the identity at exponent0 is not the order. Step identifiers and dependenciesRule: least_positive_return · Uses: G001, S043 ## Reverse checks - S044 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S043 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S042 · premise_and_result_audit — Recompute power product at k=20 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S040 · premise_and_result_audit — Recompute power product at k=19 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S038 · premise_and_result_audit — Recompute power product at k=18 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S036 · premise_and_result_audit — Recompute power product at k=17 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S034 · premise_and_result_audit — Recompute power product at k=16 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S032 · premise_and_result_audit — Recompute power product at k=15 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S030 · premise_and_result_audit — Recompute power product at k=14 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S028 · premise_and_result_audit — Recompute power product at k=13 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S026 · premise_and_result_audit — Recompute power product at k=12 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S024 · premise_and_result_audit — Recompute power product at k=11 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S022 · premise_and_result_audit — Recompute power product at k=10 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S020 · premise_and_result_audit — Recompute power product at k=9 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S018 · premise_and_result_audit — Recompute power product at k=8 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S016 · premise_and_result_audit — Recompute power product at k=7 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S014 · premise_and_result_audit — Recompute power product at k=6 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S012 · premise_and_result_audit — Recompute power product at k=5 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S010 · premise_and_result_audit — Recompute power product at k=4 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S008 · premise_and_result_audit — Recompute power product at k=3 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S006 · premise_and_result_audit — Recompute power product at k=2 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S004 · premise_and_result_audit — Recompute power product at k=1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S002 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. - S001 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi. ## Result ([[1, 2], [2, 4], [3, 8], [4, 16], [5, 7], [6, 14], [7, 3], [8, 6], [9, 12], [10, 24], [11, 23], [12, 21], [13, 17], [14, 9], [15, 18], [16, 11], [17, 22], [18, 19], [19, 13], [20, 1]], 20, Yes, Yes, order_is_the_first_positive_return) The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: d8c6bcb2e49cfac6fb35970d3b15904cded33753db1df63049e5d5f5c907b859 - candidate record SHA-256: 9012b2b83fbb62f0f6aa4080c4c814450ab42dca4f82b92d00e1c71f8e919564 - semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772 - question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0048.json)