# Question 27788: All Nonzero Prime Residues Pair with Inverses, with a Special p=2 Case [Return to catalogue page 139](https://ocme.evemisslab.com/en/questions/page/139/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261118-r04-wilson-theorem-foundations/#q-20261015-advance-20261118-r04-wilson-theorem-0038) 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. R0054-27788 · q-20261015-advance-20261118-r04-wilson-theorem-0038 ## Original question and givens Use prime modulus p=41. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence). - G001 — Use prime modulus p=41. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-wilson-theorem](https://ocme.evemisslab.com/en/mko/mko-wilson-theorem/) ## Step-by-step derivation S001 · impliesBefore source roles = [wilson-prime-inverse-pairing] After derived = [[p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence]] Conditions: none List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [41, 1] After derived = [1, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S001 S003 · impliesBefore source roles = [41, 2] After derived = [21, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S002 S004 · impliesBefore source roles = [41, 3] After derived = [14, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S003 S005 · impliesBefore source roles = [41, 4] After derived = [31, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S004 S006 · impliesBefore source roles = [41, 5] After derived = [33, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S005 S007 · impliesBefore source roles = [41, 6] After derived = [7, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S006 S008 · impliesBefore source roles = [41, 7] After derived = [6, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S007 S009 · impliesBefore source roles = [41, 8] After derived = [36, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S008 S010 · impliesBefore source roles = [41, 9] After derived = [32, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S009 S011 · impliesBefore source roles = [41, 10] After derived = [37, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S010 S012 · impliesBefore source roles = [41, 11] After derived = [15, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S011 S013 · impliesBefore source roles = [41, 12] After derived = [24, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S012 S014 · impliesBefore source roles = [41, 13] After derived = [19, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S013 S015 · impliesBefore source roles = [41, 14] After derived = [3, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S014 S016 · impliesBefore source roles = [41, 15] After derived = [11, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S015 S017 · impliesBefore source roles = [41, 16] After derived = [18, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S016 S018 · impliesBefore source roles = [41, 17] After derived = [29, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S017 S019 · impliesBefore source roles = [41, 18] After derived = [16, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S018 S020 · impliesBefore source roles = [41, 19] After derived = [13, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S019 S021 · impliesBefore source roles = [41, 20] After derived = [39, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S020 S022 · impliesBefore source roles = [41, 21] After derived = [2, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S021 S023 · impliesBefore source roles = [41, 22] After derived = [28, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S022 S024 · impliesBefore source roles = [41, 23] After derived = [25, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S023 S025 · impliesBefore source roles = [41, 24] After derived = [12, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S024 S026 · impliesBefore source roles = [41, 25] After derived = [23, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S025 S027 · impliesBefore source roles = [41, 26] After derived = [30, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S026 S028 · impliesBefore source roles = [41, 27] After derived = [38, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S027 S029 · impliesBefore source roles = [41, 28] After derived = [22, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S028 S030 · impliesBefore source roles = [41, 29] After derived = [17, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S029 S031 · impliesBefore source roles = [41, 30] After derived = [26, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S030 S032 · impliesBefore source roles = [41, 31] After derived = [4, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S031 S033 · impliesBefore source roles = [41, 32] After derived = [9, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S032 S034 · impliesBefore source roles = [41, 33] After derived = [5, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S033 S035 · impliesBefore source roles = [41, 34] After derived = [35, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S034 S036 · impliesBefore source roles = [41, 35] After derived = [34, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S035 S037 · impliesBefore source roles = [41, 36] After derived = [8, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S036 S038 · impliesBefore source roles = [41, 37] After derived = [10, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S037 S039 · impliesBefore source roles = [41, 38] After derived = [27, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S038 S040 · impliesBefore source roles = [41, 39] After derived = [20, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S039 S041 · impliesBefore source roles = [41, 40] After derived = [40, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S040 S042 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product1 := mul(1, 1) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product1 = 1 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 1, 1, 1] After derived = [0, 1, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S042 S044 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product2 := mul(1, 2) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product2 = 2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 2, 1, 2] After derived = [0, 2, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S044 S046 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product3 := mul(2, 3) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product3 = 6 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 3, 2, 6] After derived = [0, 6, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S046 S048 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product4 := mul(6, 4) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product4 = 24 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 4, 6, 24] After derived = [0, 24, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S048 S050 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 := mul(24, 5) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 = 120 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 5, 24, 120] After derived = [2, 38, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S050 S052 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(38, 6) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 228 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 6, 38, 228] After derived = [5, 23, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S052 S054 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(23, 7) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 161 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053 S055 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 7, 23, 161] After derived = [3, 38, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S054 S056 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(38, 8) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 304 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 8, 38, 304] After derived = [7, 17, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S056 S058 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(17, 9) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 153 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 9, 17, 153] After derived = [3, 30, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S058 S060 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(30, 10) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 300 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 10, 30, 300] After derived = [7, 13, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S060 S062 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(13, 11) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 143 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S061 S063 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 11, 13, 143] After derived = [3, 20, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S062 S064 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 := mul(20, 12) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 240 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S063 S065 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 12, 20, 240] After derived = [5, 35, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S064 S066 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(35, 13) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 455 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S065 S067 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 13, 35, 455] After derived = [11, 4, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S066 S068 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 := mul(4, 14) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 56 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 14, 4, 56] After derived = [1, 15, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S068 S070 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 := mul(15, 15) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 225 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069 S071 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 15, 15, 225] After derived = [5, 20, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S070 S072 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(20, 16) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 320 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 16, 20, 320] After derived = [7, 33, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S072 S074 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 := mul(33, 17) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 561 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S073 S075 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 17, 33, 561] After derived = [13, 28, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S074 S076 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(28, 18) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 504 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S075 S077 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 18, 28, 504] After derived = [12, 12, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S076 S078 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(12, 19) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 228 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S077 S079 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 19, 12, 228] After derived = [5, 23, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S078 S080 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(23, 20) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 460 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S079 S081 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 20, 23, 460] After derived = [11, 9, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S080 S082 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 := mul(9, 21) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 189 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S081 S083 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 21, 9, 189] After derived = [4, 25, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S082 S084 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 := mul(25, 22) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 550 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S083 S085 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 22, 25, 550] After derived = [13, 17, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S084 S086 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 := mul(17, 23) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 = 391 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S085 S087 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 23, 17, 391] After derived = [9, 22, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S086 S088 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 := mul(22, 24) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 = 528 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S087 S089 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 24, 22, 528] After derived = [12, 36, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S088 S090 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 := mul(36, 25) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 = 900 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S089 S091 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 25, 36, 900] After derived = [21, 39, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S090 S092 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 := mul(39, 26) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 = 1014 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S091 S093 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 26, 39, 1014] After derived = [24, 30, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S092 S094 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 := mul(30, 27) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 = 810 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S093 S095 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 27, 30, 810] After derived = [19, 31, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S094 S096 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 := mul(31, 28) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 = 868 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S095 S097 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 28, 31, 868] After derived = [21, 7, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S096 S098 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 := mul(7, 29) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 = 203 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S097 S099 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 29, 7, 203] After derived = [4, 39, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S098 S100 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 := mul(39, 30) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 = 1170 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S099 S101 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 30, 39, 1170] After derived = [28, 22, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S100 S102 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 := mul(22, 31) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 = 682 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S101 S103 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 31, 22, 682] After derived = [16, 26, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S102 S104 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 := mul(26, 32) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 = 832 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S103 S105 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 32, 26, 832] After derived = [20, 12, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S104 S106 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 := mul(12, 33) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 = 396 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S105 S107 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 33, 12, 396] After derived = [9, 27, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S106 S108 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 := mul(27, 34) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 = 918 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S107 S109 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 34, 27, 918] After derived = [22, 16, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S108 S110 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 := mul(16, 35) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 = 560 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S109 S111 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 35, 16, 560] After derived = [13, 27, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S110 S112 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 := mul(27, 36) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 = 972 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S111 S113 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 36, 27, 972] After derived = [23, 29, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S112 S114 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 := mul(29, 37) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 = 1073 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S113 S115 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 37, 29, 1073] After derived = [26, 7, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S114 S116 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 := mul(7, 38) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 = 266 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S115 S117 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 38, 7, 266] After derived = [6, 20, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S116 S118 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 := mul(20, 39) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 = 780 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S117 S119 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 39, 20, 780] After derived = [19, 1, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S118 S120 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 := mul(1, 40) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 = 40 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S119 S121 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 41, 40, 1, 40] After derived = [0, 40, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S120 S122 · impliesBefore source roles = [41, [[1, 1], [2, 21], [3, 14], [4, 31], [5, 33], [6, 7], [7, 6], [8, 36], [9, 32], [10, 37], [11, 15], [12, 24], [13, 19], [14, 3], [15, 11], [16, 18], [17, 29], [18, 16], [19, 13], [20, 39], [21, 2], [22, 28], [23, 25], [24, 12], [25, 23], [26, 30], [27, 38], [28, 22], [29, 17], [30, 26], [31, 4], [32, 9], [33, 5], [34, 35], [35, 34], [36, 8], [37, 10], [38, 27], [39, 20], [40, 40]]] After derived = [[1, 40], [[2, 21, 1], [3, 14, 1], [4, 31, 1], [5, 33, 1], [6, 7, 1], [8, 36, 1], [9, 32, 1], [10, 37, 1], [11, 15, 1], [12, 24, 1], [13, 19, 1], [16, 18, 1], [17, 29, 1], [20, 39, 1], [22, 28, 1], [23, 25, 1], [26, 30, 1], [27, 38, 1], [34, 35, 1]], 40, Yes] Conditions: none Nonself inverse pairs multiply to1.A self-inverse satisfies(r-1)(r+1)=0 modulo a prime,so is±1.For p>2 their product is-1;at p2 only the single1=-1 remains.This general teaching argument does not turn the finite table/program into a Lean proof. Step identifiers and dependenciesRule: Wilson_GENERAL_inverse_pairing_and_p2_exception · Uses: G001, S121 ## Reverse checks - S122 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S121 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S120 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 with exact rational arithmetic, not rounded decimals. - S119 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S118 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 with exact rational arithmetic, not rounded decimals. - S117 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S116 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 with exact rational arithmetic, not rounded decimals. - S115 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S114 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 with exact rational arithmetic, not rounded decimals. - S113 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S112 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 with exact rational arithmetic, not rounded decimals. - S111 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S110 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 with exact rational arithmetic, not rounded decimals. - S109 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S108 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 with exact rational arithmetic, not rounded decimals. - S107 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S106 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 with exact rational arithmetic, not rounded decimals. - S105 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S104 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 with exact rational arithmetic, not rounded decimals. - S103 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S102 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 with exact rational arithmetic, not rounded decimals. - S101 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S100 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 with exact rational arithmetic, not rounded decimals. - S099 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S098 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 with exact rational arithmetic, not rounded decimals. - S097 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S096 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 with exact rational arithmetic, not rounded decimals. - S095 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S094 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 with exact rational arithmetic, not rounded decimals. - S093 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S092 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 with exact rational arithmetic, not rounded decimals. - S091 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S090 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 with exact rational arithmetic, not rounded decimals. - S089 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S088 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 with exact rational arithmetic, not rounded decimals. - S087 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S086 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 with exact rational arithmetic, not rounded decimals. - S085 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S084 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S082 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 with exact rational arithmetic, not rounded decimals. - S081 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S080 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 with exact rational arithmetic, not rounded decimals. - S079 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S078 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 with exact rational arithmetic, not rounded decimals. - S077 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S076 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 with exact rational arithmetic, not rounded decimals. - S075 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S074 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 with exact rational arithmetic, not rounded decimals. - S073 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S072 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S070 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S068 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S066 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 with exact rational arithmetic, not rounded decimals. - S065 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S064 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S062 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S060 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S058 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S056 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S054 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S052 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S050 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S048 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product4 with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S046 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product3 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S044 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product2 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S042 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product1 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S040 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S039 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S038 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S037 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S036 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S035 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S034 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S033 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S032 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S031 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S030 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S029 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S028 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S027 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S026 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S025 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S024 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S023 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S022 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S021 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S020 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S019 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S018 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S017 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S016 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S015 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S014 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S013 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S012 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S011 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S010 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S009 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S008 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S007 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S006 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S005 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S004 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S003 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S002 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. ## Result (41, [[1, 1], [2, 21], [3, 14], [4, 31], [5, 33], [6, 7], [7, 6], [8, 36], [9, 32], [10, 37], [11, 15], [12, 24], [13, 19], [14, 3], [15, 11], [16, 18], [17, 29], [18, 16], [19, 13], [20, 39], [21, 2], [22, 28], [23, 25], [24, 12], [25, 23], [26, 30], [27, 38], [28, 22], [29, 17], [30, 26], [31, 4], [32, 9], [33, 5], [34, 35], [35, 34], [36, 8], [37, 10], [38, 27], [39, 20], [40, 40]], [1, 40], [[2, 21, 1], [3, 14, 1], [4, 31, 1], [5, 33, 1], [6, 7, 1], [8, 36, 1], [9, 32, 1], [10, 37, 1], [11, 15, 1], [12, 24, 1], [13, 19, 1], [16, 18, 1], [17, 29, 1], [20, 39, 1], [22, 28, 1], [23, 25, 1], [26, 30, 1], [27, 38, 1], [34, 35, 1]], self_inverse_residues_1_and_p_minus_1, 40, Yes) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## 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: acf49f42709a1f3f24f6a05a8aa64c782b9730f72ed6bd04843d53b5f0008ee3 - candidate record SHA-256: b61efa4ff7e833c3a8e0c98fb51098ae9da30bbe77161f05b9d7d22807a89afe - semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54 - question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261118-r04-wilson-theorem-0038.json)