# Question 20006: Euler Congruence Requires a Unit Residue, Not a Prime Modulus

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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.

R0039-20006 · q-20261014-advance-20261107-r06-euler-theorem-number-theory-0006



## Original question and givens

Use integer modulus n=10≥2 and original integer base a=7. 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=10≥2 and original integer base a=7. 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](https://ocme.evemisslab.com/en/mko/mko-euler-theorem-number-theory/)





## Step-by-step derivation

S001 · impliesBefore

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 dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · calculationBefore

Euler ORIGINAL signed base normalization exact quotient := div(7, 10)

After

Euler ORIGINAL signed base normalization exact quotient = 7/10

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 10)

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

Euler ORIGINAL signed base normalization normalized remainder := sub(7, 0)

After

Euler ORIGINAL signed base normalization normalized remainder = 7

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=7;modulus=10;exact ratio=7/10;floor quotient=0

After

value=q*n+remainder;normalized residue=7 in[0,10);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 dependenciesRule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S004





S006 · impliesBefore

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[7, 10];nonnegative initial pair=[7, 10]

After

ALL exact Euclidean rows[left,right,q,r]=[[7, 10, 0, 7], [10, 7, 1, 3], [7, 3, 2, 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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S005





S007 · impliesBefore

Euler residue representative0:ORIGINAL gcd inputs=[0, 10];nonnegative initial pair=[0, 10]

After

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

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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S006





S008 · impliesBefore

Euler residue representative1:ORIGINAL gcd inputs=[1, 10];nonnegative initial pair=[1, 10]

After

ALL exact Euclidean rows[left,right,q,r]=[[1, 10, 0, 1], [10, 1, 10, 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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S007





S009 · impliesBefore

Euler residue representative2:ORIGINAL gcd inputs=[2, 10];nonnegative initial pair=[2, 10]

After

ALL exact Euclidean rows[left,right,q,r]=[[2, 10, 0, 2], [10, 2, 5, 0]];terminal gcd=2

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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S008





S010 · impliesBefore

Euler residue representative3:ORIGINAL gcd inputs=[3, 10];nonnegative initial pair=[3, 10]

After

ALL exact Euclidean rows[left,right,q,r]=[[3, 10, 0, 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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S009





S011 · impliesBefore

Euler residue representative4:ORIGINAL gcd inputs=[4, 10];nonnegative initial pair=[4, 10]

After

ALL exact Euclidean rows[left,right,q,r]=[[4, 10, 0, 4], [10, 4, 2, 2], [4, 2, 2, 0]];terminal gcd=2

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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S010





S012 · impliesBefore

Euler residue representative5:ORIGINAL gcd inputs=[5, 10];nonnegative initial pair=[5, 10]

After

ALL exact Euclidean rows[left,right,q,r]=[[5, 10, 0, 5], [10, 5, 2, 0]];terminal gcd=5

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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S011





S013 · impliesBefore

Euler residue representative6:ORIGINAL gcd inputs=[6, 10];nonnegative initial pair=[6, 10]

After

ALL exact Euclidean rows[left,right,q,r]=[[6, 10, 0, 6], [10, 6, 1, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal gcd=2

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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S012





S014 · impliesBefore

Euler residue representative7:ORIGINAL gcd inputs=[7, 10];nonnegative initial pair=[7, 10]

After

ALL exact Euclidean rows[left,right,q,r]=[[7, 10, 0, 7], [10, 7, 1, 3], [7, 3, 2, 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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S013





S015 · impliesBefore

Euler residue representative8:ORIGINAL gcd inputs=[8, 10];nonnegative initial pair=[8, 10]

After

ALL exact Euclidean rows[left,right,q,r]=[[8, 10, 0, 8], [10, 8, 1, 2], [8, 2, 4, 0]];terminal gcd=2

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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S014





S016 · impliesBefore

Euler residue representative9:ORIGINAL gcd inputs=[9, 10];nonnegative initial pair=[9, 10]

After

ALL exact Euclidean rows[left,right,q,r]=[[9, 10, 0, 9], [10, 9, 1, 1], [9, 1, 9, 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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S015





S017 · calculationBefore

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

After

Euler complete reduced residue count phi = 4

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · impliesBefore

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

After

increasing COMPLETE U(n)=[1, 3, 7, 9];phi(n)=4;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 dependenciesRule: euler_full_units_totient_count_convention_not_prime_only · Uses: G001, S017





S019 · impliesBefore

ORIGINAL normalized base=7;COMPLETE totient exponent=4;modulus=10

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 dependenciesRule: euler_totient_power_initial_exact_invariant · Uses: G001, S018





S020 · calculationBefore

Euler power bit0 base square := mul(7, 7)

After

Euler power bit0 base square = 49

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019





S021 · calculationBefore

Euler power bit0 squared-base reduction exact quotient := div(49, 10)

After

Euler power bit0 squared-base reduction exact quotient = 49/10

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020





S022 · calculationBefore

Euler power bit0 squared-base reduction floor quotient multiple := mul(4, 10)

After

Euler power bit0 squared-base reduction floor quotient multiple = 40

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021





S023 · calculationBefore

Euler power bit0 squared-base reduction normalized remainder := sub(49, 40)

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · impliesBefore

Euler power bit0 squared-base reduction:ORIGINAL signed integer=49;modulus=10;exact ratio=49/10;floor quotient=4

After

value=q*n+remainder;normalized residue=9 in[0,10);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 dependenciesRule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S023





S025 · impliesBefore

step=0;previous[accumulator,base,exponent]=[1, 7, 4];low bit=0

After

next[accumulator,base,exponent]=[1, 9, 2];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 dependenciesRule: euler_every_binary_power_step_unit_not_required_for_arithmetic · Uses: G001, S024





S026 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S025





S027 · calculationBefore

Euler power bit1 squared-base reduction exact quotient := div(81, 10)

After

Euler power bit1 squared-base reduction exact quotient = 81/10

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026





S028 · calculationBefore

Euler power bit1 squared-base reduction floor quotient multiple := mul(8, 10)

After

Euler power bit1 squared-base reduction floor quotient multiple = 80

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027





S029 · calculationBefore

Euler power bit1 squared-base reduction normalized remainder := sub(81, 80)

After

Euler power bit1 squared-base 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S028





S030 · impliesBefore

Euler power bit1 squared-base reduction:ORIGINAL signed integer=81;modulus=10;exact ratio=81/10;floor quotient=8

After

value=q*n+remainder;normalized residue=1 in[0,10);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 dependenciesRule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S029





S031 · impliesBefore

step=1;previous[accumulator,base,exponent]=[1, 9, 2];low bit=0

After

next[accumulator,base,exponent]=[1, 1, 1];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 dependenciesRule: euler_every_binary_power_step_unit_not_required_for_arithmetic · Uses: G001, S030





S032 · calculationBefore

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

After

Euler power bit2 accumulator product = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031





S033 · calculationBefore

Euler power bit2 accumulator reduction exact quotient := div(1, 10)

After

Euler power bit2 accumulator reduction exact quotient = 1/10

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032





S034 · calculationBefore

Euler power bit2 accumulator reduction floor quotient multiple := mul(0, 10)

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S033





S035 · calculationBefore

Euler power bit2 accumulator reduction normalized remainder := sub(1, 0)

After

Euler power bit2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S034





S036 · impliesBefore

Euler power bit2 accumulator reduction:ORIGINAL signed integer=1;modulus=10;exact ratio=1/10;floor quotient=0

After

value=q*n+remainder;normalized residue=1 in[0,10);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 dependenciesRule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S035





S037 · calculationBefore

Euler power bit2 base square := mul(1, 1)

After

Euler power bit2 base square = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036





S038 · calculationBefore

Euler power bit2 squared-base reduction exact quotient := div(1, 10)

After

Euler power bit2 squared-base reduction exact quotient = 1/10

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037





S039 · calculationBefore

Euler power bit2 squared-base reduction floor quotient multiple := mul(0, 10)

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S038





S040 · calculationBefore

Euler power bit2 squared-base reduction normalized remainder := sub(1, 0)

After

Euler power bit2 squared-base 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S039





S041 · impliesBefore

Euler power bit2 squared-base reduction:ORIGINAL signed integer=1;modulus=10;exact ratio=1/10;floor quotient=0

After

value=q*n+remainder;normalized residue=1 in[0,10);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 dependenciesRule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S040





S042 · impliesBefore

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

After

next[accumulator,base,exponent]=[1, 1, 0];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 dependenciesRule: euler_every_binary_power_step_unit_not_required_for_arithmetic · Uses: G001, S041





S043 · impliesBefore

remaining exponent0;final normalized accumulator=1

After

EXACT original normalized base^4 modulo10=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 dependenciesRule: euler_totient_power_terminal_congruence_not_primality · Uses: G001, S042





S044 · calculationBefore

Euler ACTUAL base-unit inverse product := mul(7, 3)

After

Euler ACTUAL base-unit inverse product = 21

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043





S045 · calculationBefore

Euler ACTUAL inverse check exact quotient := div(21, 10)

After

Euler ACTUAL inverse check exact quotient = 21/10

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044





S046 · calculationBefore

Euler ACTUAL inverse check floor quotient multiple := mul(2, 10)

After

Euler ACTUAL inverse check floor quotient multiple = 20

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045





S047 · calculationBefore

Euler ACTUAL inverse check normalized remainder := sub(21, 20)

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S046





S048 · impliesBefore

Euler ACTUAL inverse check:ORIGINAL signed integer=21;modulus=10;exact ratio=21/10;floor quotient=2

After

value=q*n+remainder;normalized residue=1 in[0,10);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 dependenciesRule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S047





S049 · impliesBefore

ACTUAL modulus n=10;gcd(original a=7,n)=1;normalized base=7;ACTUAL inverse=3;COMPLETE U(n)=[1, 3, 7, 9]

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 dependenciesRule: euler_all_unit_residue_permutation_composite_allowed · Uses: G001, S048





S050 · impliesBefore

SAME complete U(n)=[1, 3, 7, 9] of size phi=4;multiplication by original a=7 permutes it

After

multiply ALL unit images:a^(4)*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 modulo10

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 dependenciesRule: euler_complete_unit_product_cancellation_not_nonzero_factorial · Uses: G001, S049









## Reverse checks


- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S047 · premise_and_result_audit — Recompute Euler ACTUAL inverse check normalized remainder with exact rational arithmetic, not rounded decimals.
- S046 · premise_and_result_audit — Recompute Euler ACTUAL inverse check floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S045 · premise_and_result_audit — Recompute Euler ACTUAL inverse check exact quotient with exact rational arithmetic, not rounded decimals.
- S044 · premise_and_result_audit — Recompute Euler ACTUAL base-unit inverse product with exact rational arithmetic, not rounded decimals.
- S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S040 · premise_and_result_audit — Recompute Euler power bit2 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
- S039 · premise_and_result_audit — Recompute Euler power bit2 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S038 · premise_and_result_audit — Recompute Euler power bit2 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals.
- S037 · premise_and_result_audit — Recompute Euler power bit2 base square with exact rational arithmetic, not rounded decimals.
- S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S035 · premise_and_result_audit — Recompute Euler power bit2 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals.
- S034 · premise_and_result_audit — Recompute Euler power bit2 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S033 · premise_and_result_audit — Recompute Euler power bit2 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals.
- S032 · premise_and_result_audit — Recompute Euler power bit2 accumulator product with exact rational arithmetic, not rounded decimals.
- S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S029 · premise_and_result_audit — Recompute Euler power bit1 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
- S028 · premise_and_result_audit — Recompute Euler power bit1 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S027 · premise_and_result_audit — Recompute Euler power bit1 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals.
- S026 · premise_and_result_audit — Recompute Euler power bit1 base square with exact rational arithmetic, not rounded decimals.
- S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S023 · premise_and_result_audit — Recompute Euler power bit0 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
- S022 · premise_and_result_audit — Recompute Euler power bit0 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S021 · premise_and_result_audit — Recompute Euler power bit0 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals.
- S020 · premise_and_result_audit — Recompute Euler power bit0 base square 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 — Recompute Euler complete reduced residue count phi with exact rational arithmetic, not rounded decimals.
- 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

(10, 7, 4, 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: 3da730110053d84d34a660f8a9192a09c3d77fe14a71a9e936b86bf8810ec4a8
- candidate record SHA-256: 06faf3b92ab8f62a89f55a46000f28d8aa6dad351d250a180169cdfa250dbe25
- semantic review receipt SHA-256: 6e11e35fafe32f16900ce9b522d01acb7b2ee6b463ab179392076cfbe1af8034
- question-first witness SHA-256: cc0fa361fc25151729e70264340d2eae94f179f6f2a2edfe80ba5704fbbd0682
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0006.json)