Question 20009: Euler Congruence Requires a Unit Residue, Not a Prime Modulus

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0039-20009 · q-20261014-advance-20261107-r06-euler-theorem-number-theory-0009

Original question and givens

Use integer modulus n=13≥2 and original integer base a=10. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

G001 — Use integer modulus n=13≥2 and original integer base a=10. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

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-euler-theorem-number-theory

Step-by-step derivation

S001 · implies

Before

source operation=eulerthm-congruence;original snapshot and mathematics retained

After

explicit ordered source output fields=[n, normalized_base, totient, residue, congruent_to_one]

Conditions: none

Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source claim scope.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · calculation

Before

Euler ORIGINAL signed base normalization exact quotient := div(10, 13)

After

Euler ORIGINAL signed base normalization exact quotient = 10/13

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 13)

After

Euler ORIGINAL signed base normalization floor quotient multiple = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Euler ORIGINAL signed base normalization normalized remainder := sub(10, 0)

After

Euler ORIGINAL signed base normalization normalized remainder = 10

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=10;modulus=13;exact ratio=10/13;floor quotient=0

After

value=q*n+remainder;normalized residue=10 in[0,13);same congruence class,NOT absolute-value replacement

Conditions: none

Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S004

S006 · implies

Before

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[10, 13];nonnegative initial pair=[10, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[10, 13, 0, 10], [13, 10, 1, 3], [10, 3, 3, 1], [3, 1, 3, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S005

S007 · implies

Before

Euler residue representative0:ORIGINAL gcd inputs=[0, 13];nonnegative initial pair=[0, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[0, 13, 0, 0]];terminal gcd=13

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S006

S008 · implies

Before

Euler residue representative1:ORIGINAL gcd inputs=[1, 13];nonnegative initial pair=[1, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[1, 13, 0, 1], [13, 1, 13, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S007

S009 · implies

Before

Euler residue representative2:ORIGINAL gcd inputs=[2, 13];nonnegative initial pair=[2, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[2, 13, 0, 2], [13, 2, 6, 1], [2, 1, 2, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S008

S010 · implies

Before

Euler residue representative3:ORIGINAL gcd inputs=[3, 13];nonnegative initial pair=[3, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[3, 13, 0, 3], [13, 3, 4, 1], [3, 1, 3, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S009

S011 · implies

Before

Euler residue representative4:ORIGINAL gcd inputs=[4, 13];nonnegative initial pair=[4, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[4, 13, 0, 4], [13, 4, 3, 1], [4, 1, 4, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S010

S012 · implies

Before

Euler residue representative5:ORIGINAL gcd inputs=[5, 13];nonnegative initial pair=[5, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[5, 13, 0, 5], [13, 5, 2, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S011

S013 · implies

Before

Euler residue representative6:ORIGINAL gcd inputs=[6, 13];nonnegative initial pair=[6, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[6, 13, 0, 6], [13, 6, 2, 1], [6, 1, 6, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S012

S014 · implies

Before

Euler residue representative7:ORIGINAL gcd inputs=[7, 13];nonnegative initial pair=[7, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[7, 13, 0, 7], [13, 7, 1, 6], [7, 6, 1, 1], [6, 1, 6, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S013

S015 · implies

Before

Euler residue representative8:ORIGINAL gcd inputs=[8, 13];nonnegative initial pair=[8, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[8, 13, 0, 8], [13, 8, 1, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S014

S016 · implies

Before

Euler residue representative9:ORIGINAL gcd inputs=[9, 13];nonnegative initial pair=[9, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[9, 13, 0, 9], [13, 9, 1, 4], [9, 4, 2, 1], [4, 1, 4, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S015

S017 · implies

Before

Euler residue representative10:ORIGINAL gcd inputs=[10, 13];nonnegative initial pair=[10, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[10, 13, 0, 10], [13, 10, 1, 3], [10, 3, 3, 1], [3, 1, 3, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S016

S018 · implies

Before

Euler residue representative11:ORIGINAL gcd inputs=[11, 13];nonnegative initial pair=[11, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[11, 13, 0, 11], [13, 11, 1, 2], [11, 2, 5, 1], [2, 1, 2, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S017

S019 · implies

Before

Euler residue representative12:ORIGINAL gcd inputs=[12, 13];nonnegative initial pair=[12, 13]

After

ALL exact Euclidean rows[left,right,q,r]=[[12, 13, 0, 12], [13, 12, 1, 1], [12, 1, 12, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S018

S020 · calculation

Before

Euler complete reduced residue count phi := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

After

Euler complete reduced residue count phi = 12

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

COMPLETE original residue carrier0..12;ALL [representative,gcd,unit] rows=[[0, 13, No], [1, 1, Yes], [2, 1, Yes], [3, 1, Yes], [4, 1, Yes], [5, 1, Yes], [6, 1, Yes], [7, 1, Yes], [8, 1, Yes], [9, 1, Yes], [10, 1, Yes], [11, 1, Yes], [12, 1, Yes]]

After

increasing COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];phi(n)=12;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

Conditions: none

Test gcd1 for every representative to obtain the complete unit set,which contains1.The source totient count uses1..n while the unit list uses0..n−1.For n≥2 both swapped endpoints0 and n are nonunits,so the counts agree without assuming a prime modulus.

Step identifiers and dependencies

Rule: euler_full_units_totient_count_convention_not_prime_only · Uses: G001, S020

S022 · implies

Before

ORIGINAL normalized base=10;COMPLETE totient exponent=12;modulus=13

After

initialize accumulator1,current_base=base,remaining=phi;invariant accumulator*current_base^remaining congruent to the original totient power

Conditions: none

Use finite square-and-multiply steps preserving the original totient-power congruence,without constructing huge ordinary powers.Phi is the complete unit count,not n−1 unless primality is separately established.

Step identifiers and dependencies

Rule: euler_totient_power_initial_exact_invariant · Uses: G001, S021

S023 · calculation

Before

Euler power bit0 base square := mul(10, 10)

After

Euler power bit0 base square = 100

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

Euler power bit0 squared-base reduction exact quotient := div(100, 13)

After

Euler power bit0 squared-base reduction exact quotient = 100/13

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

Euler power bit0 squared-base reduction floor quotient multiple := mul(7, 13)

After

Euler power bit0 squared-base reduction floor quotient multiple = 91

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

Euler power bit0 squared-base reduction normalized remainder := sub(100, 91)

After

Euler power bit0 squared-base reduction normalized remainder = 9

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · implies

Before

Euler power bit0 squared-base reduction:ORIGINAL signed integer=100;modulus=13;exact ratio=100/13;floor quotient=7

After

value=q*n+remainder;normalized residue=9 in[0,13);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S026

S028 · implies

Before

step=0;previous[accumulator,base,exponent]=[1, 10, 12];low bit=0

After

next[accumulator,base,exponent]=[1, 9, 6];split exponent=2*q+bit preserves original-power invariant

Conditions: none

For an odd power move one base factor into the accumulator and square the base for the even part.Exact reductions preserve congruence and halving guarantees termination.This arithmetic also works for nonunits;computability does not establish Euler applicability.

Step identifiers and dependencies

Rule: euler_every_binary_power_step_unit_not_required_for_arithmetic · Uses: G001, S027

S029 · calculation

Before

Euler power bit1 base square := mul(9, 9)

After

Euler power bit1 base square = 81

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

Euler power bit1 squared-base reduction exact quotient := div(81, 13)

After

Euler power bit1 squared-base reduction exact quotient = 81/13

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

Euler power bit1 squared-base reduction floor quotient multiple := mul(6, 13)

After

Euler power bit1 squared-base reduction floor quotient multiple = 78

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

Euler power bit1 squared-base reduction normalized remainder := sub(81, 78)

After

Euler power bit1 squared-base reduction normalized remainder = 3

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · implies

Before

Euler power bit1 squared-base reduction:ORIGINAL signed integer=81;modulus=13;exact ratio=81/13;floor quotient=6

After

value=q*n+remainder;normalized residue=3 in[0,13);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S032

S034 · implies

Before

step=1;previous[accumulator,base,exponent]=[1, 9, 6];low bit=0

After

next[accumulator,base,exponent]=[1, 3, 3];split exponent=2*q+bit preserves original-power invariant

Conditions: none

Step identifiers and dependencies

Rule: euler_every_binary_power_step_unit_not_required_for_arithmetic · Uses: G001, S033

S035 · calculation

Before

Euler power bit2 accumulator product := mul(1, 3)

After

Euler power bit2 accumulator product = 3

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

Euler power bit2 accumulator reduction exact quotient := div(3, 13)

After

Euler power bit2 accumulator reduction exact quotient = 3/13

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

Euler power bit2 accumulator reduction floor quotient multiple := mul(0, 13)

After

Euler power bit2 accumulator reduction floor quotient multiple = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

Euler power bit2 accumulator reduction normalized remainder := sub(3, 0)

After

Euler power bit2 accumulator reduction normalized remainder = 3

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · implies

Before

Euler power bit2 accumulator reduction:ORIGINAL signed integer=3;modulus=13;exact ratio=3/13;floor quotient=0

After

value=q*n+remainder;normalized residue=3 in[0,13);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S038

S040 · calculation

Before

Euler power bit2 base square := mul(3, 3)

After

Euler power bit2 base square = 9

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

Euler power bit2 squared-base reduction exact quotient := div(9, 13)

After

Euler power bit2 squared-base reduction exact quotient = 9/13

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

Euler power bit2 squared-base reduction floor quotient multiple := mul(0, 13)

After

Euler power bit2 squared-base reduction floor quotient multiple = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

Euler power bit2 squared-base reduction normalized remainder := sub(9, 0)

After

Euler power bit2 squared-base reduction normalized remainder = 9

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · implies

Before

Euler power bit2 squared-base reduction:ORIGINAL signed integer=9;modulus=13;exact ratio=9/13;floor quotient=0

After

value=q*n+remainder;normalized residue=9 in[0,13);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S043

S045 · implies

Before

step=2;previous[accumulator,base,exponent]=[1, 3, 3];low bit=1

After

next[accumulator,base,exponent]=[3, 9, 1];split exponent=2*q+bit preserves original-power invariant

Conditions: none

Step identifiers and dependencies

Rule: euler_every_binary_power_step_unit_not_required_for_arithmetic · Uses: G001, S044

S046 · calculation

Before

Euler power bit3 accumulator product := mul(3, 9)

After

Euler power bit3 accumulator product = 27

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · calculation

Before

Euler power bit3 accumulator reduction exact quotient := div(27, 13)

After

Euler power bit3 accumulator reduction exact quotient = 27/13

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · calculation

Before

Euler power bit3 accumulator reduction floor quotient multiple := mul(2, 13)

After

Euler power bit3 accumulator reduction floor quotient multiple = 26

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

Euler power bit3 accumulator reduction normalized remainder := sub(27, 26)

After

Euler power bit3 accumulator reduction normalized remainder = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · implies

Before

Euler power bit3 accumulator reduction:ORIGINAL signed integer=27;modulus=13;exact ratio=27/13;floor quotient=2

After

value=q*n+remainder;normalized residue=1 in[0,13);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S049

S051 · calculation

Before

Euler power bit3 base square := mul(9, 9)

After

Euler power bit3 base square = 81

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · calculation

Before

Euler power bit3 squared-base reduction exact quotient := div(81, 13)

After

Euler power bit3 squared-base reduction exact quotient = 81/13

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · calculation

Before

Euler power bit3 squared-base reduction floor quotient multiple := mul(6, 13)

After

Euler power bit3 squared-base reduction floor quotient multiple = 78

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · calculation

Before

Euler power bit3 squared-base reduction normalized remainder := sub(81, 78)

After

Euler power bit3 squared-base reduction normalized remainder = 3

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · implies

Before

Euler power bit3 squared-base reduction:ORIGINAL signed integer=81;modulus=13;exact ratio=81/13;floor quotient=6

After

value=q*n+remainder;normalized residue=3 in[0,13);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S054

S056 · implies

Before

step=3;previous[accumulator,base,exponent]=[3, 9, 1];low bit=1

After

next[accumulator,base,exponent]=[1, 3, 0];split exponent=2*q+bit preserves original-power invariant

Conditions: none

Step identifiers and dependencies

Rule: euler_every_binary_power_step_unit_not_required_for_arithmetic · Uses: G001, S055

S057 · implies

Before

remaining exponent0;final normalized accumulator=1

After

EXACT original normalized base^12 modulo13=1

Conditions: none

The terminal invariant gives the original totient-power residue.This is arithmetic modulo the stated modulus,not a primality certificate.

Step identifiers and dependencies

Rule: euler_totient_power_terminal_congruence_not_primality · Uses: G001, S056

S058 · calculation

Before

Euler ACTUAL base-unit inverse product := mul(10, 4)

After

Euler ACTUAL base-unit inverse product = 40

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · calculation

Before

Euler ACTUAL inverse check exact quotient := div(40, 13)

After

Euler ACTUAL inverse check exact quotient = 40/13

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · calculation

Before

Euler ACTUAL inverse check floor quotient multiple := mul(3, 13)

After

Euler ACTUAL inverse check floor quotient multiple = 39

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · calculation

Before

Euler ACTUAL inverse check normalized remainder := sub(40, 39)

After

Euler ACTUAL inverse check normalized remainder = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · implies

Before

Euler ACTUAL inverse check:ORIGINAL signed integer=40;modulus=13;exact ratio=40/13;floor quotient=3

After

value=q*n+remainder;normalized residue=1 in[0,13);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S061

S063 · implies

Before

ACTUAL modulus n=13;gcd(original a=10,n)=1;normalized base=10;ACTUAL inverse=4;COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

After

for ANY r in U(n),a*r remains a unit;actual inverse makes multiplication injective;finite self-injection on U(n) is surjective,so a*U(n)=U(n)

Conditions: none

The reverse product of inverses inverts a product of units,so multiplication by a stays in U(n).Its actual inverse proves injectivity and a finite self-injection is surjective.The modulus may be composite;not every nonzero residue must be a unit.

Step identifiers and dependencies

Rule: euler_all_unit_residue_permutation_composite_allowed · Uses: G001, S062

S064 · implies

Before

SAME complete U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] of size phi=12;multiplication by original a=10 permutes it

After

multiply ALL unit images:a^(12)*product(U) congruent product(U);EVERY U factor is a UNIT,so product(U) has reverse-product inverse;cancel ONLY this unit to get a^phi congruent1 modulo13

Conditions: none

Products over the same permuted finite unit set agree.Cancel product(U) only because it is a unit,not because it is merely nonzero.This is not cancellation of all nonzero residues under a composite modulus or cancellation of0.No prime premise is used,and a passing Euler congruence supplies no primality converse.

Step identifiers and dependencies

Rule: euler_complete_unit_product_cancellation_not_nonzero_factorial · Uses: G001, S063

Reverse checks

S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S061 · premise_and_result_audit — Recompute Euler ACTUAL inverse check normalized remainder with exact rational arithmetic, not rounded decimals.
S060 · premise_and_result_audit — Recompute Euler ACTUAL inverse check floor quotient multiple with exact rational arithmetic, not rounded decimals.
S059 · premise_and_result_audit — Recompute Euler ACTUAL inverse check exact quotient with exact rational arithmetic, not rounded decimals.
S058 · premise_and_result_audit — Recompute Euler ACTUAL base-unit inverse product with exact rational arithmetic, not rounded decimals.
S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S054 · premise_and_result_audit — Recompute Euler power bit3 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S053 · premise_and_result_audit — Recompute Euler power bit3 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S052 · premise_and_result_audit — Recompute Euler power bit3 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S051 · premise_and_result_audit — Recompute Euler power bit3 base square with exact rational arithmetic, not rounded decimals.
S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S049 · premise_and_result_audit — Recompute Euler power bit3 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S048 · premise_and_result_audit — Recompute Euler power bit3 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S047 · premise_and_result_audit — Recompute Euler power bit3 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals.
S046 · premise_and_result_audit — Recompute Euler power bit3 accumulator product with exact rational arithmetic, not rounded decimals.
S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S043 · premise_and_result_audit — Recompute Euler power bit2 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Recompute Euler power bit2 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S041 · premise_and_result_audit — Recompute Euler power bit2 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Recompute Euler power bit2 base square with exact rational arithmetic, not rounded decimals.
S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S038 · premise_and_result_audit — Recompute Euler power bit2 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S037 · premise_and_result_audit — Recompute Euler power bit2 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Recompute Euler power bit2 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals.
S035 · premise_and_result_audit — Recompute Euler power bit2 accumulator product with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S032 · premise_and_result_audit — Recompute Euler power bit1 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S031 · premise_and_result_audit — Recompute Euler power bit1 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S030 · premise_and_result_audit — Recompute Euler power bit1 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Recompute Euler power bit1 base square with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S026 · premise_and_result_audit — Recompute Euler power bit0 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute Euler power bit0 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute Euler power bit0 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute Euler power bit0 base square with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S020 · premise_and_result_audit — Recompute Euler complete reduced residue count phi with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S004 · premise_and_result_audit — Recompute Euler ORIGINAL signed base normalization normalized remainder with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute Euler ORIGINAL signed base normalization floor quotient multiple with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute Euler ORIGINAL signed base normalization exact quotient with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(13, 10, 12, 1, 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: 6705fe0affde78b7222c9fdc094a172f0fdb72e717b31b1a15524c5bbd7d44e9
candidate record SHA-256: 5d2108c9bf8a06fe33796bd9ddc501333c322937001ad21717763d02dacf378c
semantic review receipt SHA-256: 6e11e35fafe32f16900ce9b522d01acb7b2ee6b463ab179392076cfbe1af8034
question-first witness SHA-256: cc0fa361fc25151729e70264340d2eae94f179f6f2a2edfe80ba5704fbbd0682
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON