# Question 20092: Multiplication by a Unit Permutes the Complete Reduced Residues

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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.

R0039-20092 · q-20261014-advance-20261107-r06-euler-theorem-number-theory-0042



## Original question and givens

Use integer modulus n=21≥2 and original integer base a=19. 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 (units, sorted_mapped_units, permutation, certificate).


- G001 — Use integer modulus n=21≥2 and original integer base a=19. 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 (units, sorted_mapped_units, permutation, certificate).





## 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-unit-permutation;original snapshot and mathematics retained

After

explicit ordered source output fields=[units, sorted_mapped_units, permutation, certificate]

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(19, 21)

After

Euler ORIGINAL signed base normalization exact quotient = 19/21

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, S001





S003 · calculationBefore

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 21)

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(19, 0)

After

Euler ORIGINAL signed base normalization normalized remainder = 19

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=19;modulus=21;exact ratio=19/21;floor quotient=0

After

value=q*n+remainder;normalized residue=19 in[0,21);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=[19, 21];nonnegative initial pair=[19, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[19, 21, 0, 19], [21, 19, 1, 2], [19, 2, 9, 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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S005





S007 · impliesBefore

Euler residue representative0:ORIGINAL gcd inputs=[0, 21];nonnegative initial pair=[0, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[0, 21, 0, 0]];terminal gcd=21

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, 21];nonnegative initial pair=[1, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[1, 21, 0, 1], [21, 1, 21, 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, 21];nonnegative initial pair=[2, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[2, 21, 0, 2], [21, 2, 10, 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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S008





S010 · impliesBefore

Euler residue representative3:ORIGINAL gcd inputs=[3, 21];nonnegative initial pair=[3, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[3, 21, 0, 3], [21, 3, 7, 0]];terminal gcd=3

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, 21];nonnegative initial pair=[4, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[4, 21, 0, 4], [21, 4, 5, 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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S010





S012 · impliesBefore

Euler residue representative5:ORIGINAL gcd inputs=[5, 21];nonnegative initial pair=[5, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[5, 21, 0, 5], [21, 5, 4, 1], [5, 1, 5, 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, S011





S013 · impliesBefore

Euler residue representative6:ORIGINAL gcd inputs=[6, 21];nonnegative initial pair=[6, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[6, 21, 0, 6], [21, 6, 3, 3], [6, 3, 2, 0]];terminal gcd=3

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, 21];nonnegative initial pair=[7, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[7, 21, 0, 7], [21, 7, 3, 0]];terminal gcd=7

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, 21];nonnegative initial pair=[8, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[8, 21, 0, 8], [21, 8, 2, 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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S014





S016 · impliesBefore

Euler residue representative9:ORIGINAL gcd inputs=[9, 21];nonnegative initial pair=[9, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[9, 21, 0, 9], [21, 9, 2, 3], [9, 3, 3, 0]];terminal gcd=3

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 · impliesBefore

Euler residue representative10:ORIGINAL gcd inputs=[10, 21];nonnegative initial pair=[10, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[10, 21, 0, 10], [21, 10, 2, 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, S016





S018 · impliesBefore

Euler residue representative11:ORIGINAL gcd inputs=[11, 21];nonnegative initial pair=[11, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[11, 21, 0, 11], [21, 11, 1, 10], [11, 10, 1, 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, S017





S019 · impliesBefore

Euler residue representative12:ORIGINAL gcd inputs=[12, 21];nonnegative initial pair=[12, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[12, 21, 0, 12], [21, 12, 1, 9], [12, 9, 1, 3], [9, 3, 3, 0]];terminal gcd=3

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, S018





S020 · impliesBefore

Euler residue representative13:ORIGINAL gcd inputs=[13, 21];nonnegative initial pair=[13, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[13, 21, 0, 13], [21, 13, 1, 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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S019





S021 · impliesBefore

Euler residue representative14:ORIGINAL gcd inputs=[14, 21];nonnegative initial pair=[14, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[14, 21, 0, 14], [21, 14, 1, 7], [14, 7, 2, 0]];terminal gcd=7

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, S020





S022 · impliesBefore

Euler residue representative15:ORIGINAL gcd inputs=[15, 21];nonnegative initial pair=[15, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[15, 21, 0, 15], [21, 15, 1, 6], [15, 6, 2, 3], [6, 3, 2, 0]];terminal gcd=3

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, S021





S023 · impliesBefore

Euler residue representative16:ORIGINAL gcd inputs=[16, 21];nonnegative initial pair=[16, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[16, 21, 0, 16], [21, 16, 1, 5], [16, 5, 3, 1], [5, 1, 5, 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, S022





S024 · impliesBefore

Euler residue representative17:ORIGINAL gcd inputs=[17, 21];nonnegative initial pair=[17, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[17, 21, 0, 17], [21, 17, 1, 4], [17, 4, 4, 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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S023





S025 · impliesBefore

Euler residue representative18:ORIGINAL gcd inputs=[18, 21];nonnegative initial pair=[18, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[18, 21, 0, 18], [21, 18, 1, 3], [18, 3, 6, 0]];terminal gcd=3

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, S024





S026 · impliesBefore

Euler residue representative19:ORIGINAL gcd inputs=[19, 21];nonnegative initial pair=[19, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[19, 21, 0, 19], [21, 19, 1, 2], [19, 2, 9, 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 dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S025





S027 · impliesBefore

Euler residue representative20:ORIGINAL gcd inputs=[20, 21];nonnegative initial pair=[20, 21]

After

ALL exact Euclidean rows[left,right,q,r]=[[20, 21, 0, 20], [21, 20, 1, 1], [20, 1, 20, 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, S026





S028 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S027





S029 · impliesBefore

COMPLETE original residue carrier0..20;ALL [representative,gcd,unit] rows=[[0, 21, No], [1, 1, Yes], [2, 1, Yes], [3, 3, No], [4, 1, Yes], [5, 1, Yes], [6, 3, No], [7, 7, No], [8, 1, Yes], [9, 3, No], [10, 1, Yes], [11, 1, Yes], [12, 3, No], [13, 1, Yes], [14, 7, No], [15, 3, No], [16, 1, Yes], [17, 1, Yes], [18, 3, No], [19, 1, Yes], [20, 1, Yes]]

After

increasing COMPLETE U(n)=[1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20];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 dependenciesRule: euler_full_units_totient_count_convention_not_prime_only · Uses: G001, S028





S030 · calculationBefore

Euler ACTUAL base-unit inverse product := mul(19, 10)

After

Euler ACTUAL base-unit inverse product = 190

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, S029





S031 · calculationBefore

Euler ACTUAL inverse check exact quotient := div(190, 21)

After

Euler ACTUAL inverse check exact quotient = 190/21

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, S030





S032 · calculationBefore

Euler ACTUAL inverse check floor quotient multiple := mul(9, 21)

After

Euler ACTUAL inverse check floor quotient multiple = 189

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 ACTUAL inverse check normalized remainder := sub(190, 189)

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, S032





S034 · impliesBefore

Euler ACTUAL inverse check:ORIGINAL signed integer=190;modulus=21;exact ratio=190/21;floor quotient=9

After

value=q*n+remainder;normalized residue=1 in[0,21);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, S033





S035 · impliesBefore

ACTUAL modulus n=21;gcd(original a=19,n)=1;normalized base=19;ACTUAL inverse=10;COMPLETE U(n)=[1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20]

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, S034





S036 · calculationBefore

Euler ORIGINAL a times unit representative1 := mul(19, 1)

After

Euler ORIGINAL a times unit representative1 = 19

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, S035





S037 · calculationBefore

Euler unit representative1 raw image exact quotient := div(19, 21)

After

Euler unit representative1 raw image exact quotient = 19/21

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, S036





S038 · calculationBefore

Euler unit representative1 raw image floor quotient multiple := mul(0, 21)

After

Euler unit representative1 raw image 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, S037





S039 · calculationBefore

Euler unit representative1 raw image normalized remainder := sub(19, 0)

After

Euler unit representative1 raw image normalized remainder = 19

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 · impliesBefore

Euler unit representative1 raw image:ORIGINAL signed integer=19;modulus=21;exact ratio=19/21;floor quotient=0

After

value=q*n+remainder;normalized residue=19 in[0,21);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, S039





S041 · calculationBefore

Euler ORIGINAL a times unit representative2 := mul(19, 2)

After

Euler ORIGINAL a times unit representative2 = 38

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, S040





S042 · calculationBefore

Euler unit representative2 raw image exact quotient := div(38, 21)

After

Euler unit representative2 raw image exact quotient = 38/21

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, S041





S043 · calculationBefore

Euler unit representative2 raw image floor quotient multiple := mul(1, 21)

After

Euler unit representative2 raw image floor quotient multiple = 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, S042





S044 · calculationBefore

Euler unit representative2 raw image normalized remainder := sub(38, 21)

After

Euler unit representative2 raw image normalized remainder = 17

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 · impliesBefore

Euler unit representative2 raw image:ORIGINAL signed integer=38;modulus=21;exact ratio=38/21;floor quotient=1

After

value=q*n+remainder;normalized residue=17 in[0,21);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, S044





S046 · calculationBefore

Euler ORIGINAL a times unit representative4 := mul(19, 4)

After

Euler ORIGINAL a times unit representative4 = 76

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 unit representative4 raw image exact quotient := div(76, 21)

After

Euler unit representative4 raw image exact quotient = 76/21

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, S046





S048 · calculationBefore

Euler unit representative4 raw image floor quotient multiple := mul(3, 21)

After

Euler unit representative4 raw image floor quotient multiple = 63

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, S047





S049 · calculationBefore

Euler unit representative4 raw image normalized remainder := sub(76, 63)

After

Euler unit representative4 raw image normalized remainder = 13

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, S048





S050 · impliesBefore

Euler unit representative4 raw image:ORIGINAL signed integer=76;modulus=21;exact ratio=76/21;floor quotient=3

After

value=q*n+remainder;normalized residue=13 in[0,21);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, S049





S051 · calculationBefore

Euler ORIGINAL a times unit representative5 := mul(19, 5)

After

Euler ORIGINAL a times unit representative5 = 95

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, S050





S052 · calculationBefore

Euler unit representative5 raw image exact quotient := div(95, 21)

After

Euler unit representative5 raw image exact quotient = 95/21

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, S051





S053 · calculationBefore

Euler unit representative5 raw image floor quotient multiple := mul(4, 21)

After

Euler unit representative5 raw image floor quotient multiple = 84

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, S052





S054 · calculationBefore

Euler unit representative5 raw image normalized remainder := sub(95, 84)

After

Euler unit representative5 raw image normalized remainder = 11

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, S053





S055 · impliesBefore

Euler unit representative5 raw image:ORIGINAL signed integer=95;modulus=21;exact ratio=95/21;floor quotient=4

After

value=q*n+remainder;normalized residue=11 in[0,21);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, S054





S056 · calculationBefore

Euler ORIGINAL a times unit representative8 := mul(19, 8)

After

Euler ORIGINAL a times unit representative8 = 152

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, S055





S057 · calculationBefore

Euler unit representative8 raw image exact quotient := div(152, 21)

After

Euler unit representative8 raw image exact quotient = 152/21

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, S056





S058 · calculationBefore

Euler unit representative8 raw image floor quotient multiple := mul(7, 21)

After

Euler unit representative8 raw image floor quotient multiple = 147

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, S057





S059 · calculationBefore

Euler unit representative8 raw image normalized remainder := sub(152, 147)

After

Euler unit representative8 raw image normalized remainder = 5

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, S058





S060 · impliesBefore

Euler unit representative8 raw image:ORIGINAL signed integer=152;modulus=21;exact ratio=152/21;floor quotient=7

After

value=q*n+remainder;normalized residue=5 in[0,21);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, S059





S061 · calculationBefore

Euler ORIGINAL a times unit representative10 := mul(19, 10)

After

Euler ORIGINAL a times unit representative10 = 190

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, S060





S062 · calculationBefore

Euler unit representative10 raw image exact quotient := div(190, 21)

After

Euler unit representative10 raw image exact quotient = 190/21

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, S061





S063 · calculationBefore

Euler unit representative10 raw image floor quotient multiple := mul(9, 21)

After

Euler unit representative10 raw image floor quotient multiple = 189

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, S062





S064 · calculationBefore

Euler unit representative10 raw image normalized remainder := sub(190, 189)

After

Euler unit representative10 raw image 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, S063





S065 · impliesBefore

Euler unit representative10 raw image:ORIGINAL signed integer=190;modulus=21;exact ratio=190/21;floor quotient=9

After

value=q*n+remainder;normalized residue=1 in[0,21);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, S064





S066 · calculationBefore

Euler ORIGINAL a times unit representative11 := mul(19, 11)

After

Euler ORIGINAL a times unit representative11 = 209

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, S065





S067 · calculationBefore

Euler unit representative11 raw image exact quotient := div(209, 21)

After

Euler unit representative11 raw image exact quotient = 209/21

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, S066





S068 · calculationBefore

Euler unit representative11 raw image floor quotient multiple := mul(9, 21)

After

Euler unit representative11 raw image floor quotient multiple = 189

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, S067





S069 · calculationBefore

Euler unit representative11 raw image normalized remainder := sub(209, 189)

After

Euler unit representative11 raw image normalized remainder = 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, S068





S070 · impliesBefore

Euler unit representative11 raw image:ORIGINAL signed integer=209;modulus=21;exact ratio=209/21;floor quotient=9

After

value=q*n+remainder;normalized residue=20 in[0,21);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, S069





S071 · calculationBefore

Euler ORIGINAL a times unit representative13 := mul(19, 13)

After

Euler ORIGINAL a times unit representative13 = 247

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, S070





S072 · calculationBefore

Euler unit representative13 raw image exact quotient := div(247, 21)

After

Euler unit representative13 raw image exact quotient = 247/21

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, S071





S073 · calculationBefore

Euler unit representative13 raw image floor quotient multiple := mul(11, 21)

After

Euler unit representative13 raw image floor quotient multiple = 231

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, S072





S074 · calculationBefore

Euler unit representative13 raw image normalized remainder := sub(247, 231)

After

Euler unit representative13 raw image normalized remainder = 16

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, S073





S075 · impliesBefore

Euler unit representative13 raw image:ORIGINAL signed integer=247;modulus=21;exact ratio=247/21;floor quotient=11

After

value=q*n+remainder;normalized residue=16 in[0,21);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, S074





S076 · calculationBefore

Euler ORIGINAL a times unit representative16 := mul(19, 16)

After

Euler ORIGINAL a times unit representative16 = 304

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, S075





S077 · calculationBefore

Euler unit representative16 raw image exact quotient := div(304, 21)

After

Euler unit representative16 raw image exact quotient = 304/21

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, S076





S078 · calculationBefore

Euler unit representative16 raw image floor quotient multiple := mul(14, 21)

After

Euler unit representative16 raw image floor quotient multiple = 294

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, S077





S079 · calculationBefore

Euler unit representative16 raw image normalized remainder := sub(304, 294)

After

Euler unit representative16 raw image 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S078





S080 · impliesBefore

Euler unit representative16 raw image:ORIGINAL signed integer=304;modulus=21;exact ratio=304/21;floor quotient=14

After

value=q*n+remainder;normalized residue=10 in[0,21);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, S079





S081 · calculationBefore

Euler ORIGINAL a times unit representative17 := mul(19, 17)

After

Euler ORIGINAL a times unit representative17 = 323

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, S080





S082 · calculationBefore

Euler unit representative17 raw image exact quotient := div(323, 21)

After

Euler unit representative17 raw image exact quotient = 323/21

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, S081





S083 · calculationBefore

Euler unit representative17 raw image floor quotient multiple := mul(15, 21)

After

Euler unit representative17 raw image floor quotient multiple = 315

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, S082





S084 · calculationBefore

Euler unit representative17 raw image normalized remainder := sub(323, 315)

After

Euler unit representative17 raw image normalized remainder = 8

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, S083





S085 · impliesBefore

Euler unit representative17 raw image:ORIGINAL signed integer=323;modulus=21;exact ratio=323/21;floor quotient=15

After

value=q*n+remainder;normalized residue=8 in[0,21);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, S084





S086 · calculationBefore

Euler ORIGINAL a times unit representative19 := mul(19, 19)

After

Euler ORIGINAL a times unit representative19 = 361

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, S085





S087 · calculationBefore

Euler unit representative19 raw image exact quotient := div(361, 21)

After

Euler unit representative19 raw image exact quotient = 361/21

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, S086





S088 · calculationBefore

Euler unit representative19 raw image floor quotient multiple := mul(17, 21)

After

Euler unit representative19 raw image floor quotient multiple = 357

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, S087





S089 · calculationBefore

Euler unit representative19 raw image normalized remainder := sub(361, 357)

After

Euler unit representative19 raw image normalized remainder = 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, S088





S090 · impliesBefore

Euler unit representative19 raw image:ORIGINAL signed integer=361;modulus=21;exact ratio=361/21;floor quotient=17

After

value=q*n+remainder;normalized residue=4 in[0,21);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, S089





S091 · calculationBefore

Euler ORIGINAL a times unit representative20 := mul(19, 20)

After

Euler ORIGINAL a times unit representative20 = 380

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, S090





S092 · calculationBefore

Euler unit representative20 raw image exact quotient := div(380, 21)

After

Euler unit representative20 raw image exact quotient = 380/21

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, S091





S093 · calculationBefore

Euler unit representative20 raw image floor quotient multiple := mul(18, 21)

After

Euler unit representative20 raw image floor quotient multiple = 378

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, S092





S094 · calculationBefore

Euler unit representative20 raw image normalized remainder := sub(380, 378)

After

Euler unit representative20 raw image normalized remainder = 2

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, S093





S095 · impliesBefore

Euler unit representative20 raw image:ORIGINAL signed integer=380;modulus=21;exact ratio=380/21;floor quotient=18

After

value=q*n+remainder;normalized residue=2 in[0,21);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, S094





S096 · impliesBefore

COMPLETE original increasing U(n)=[1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20];actual images in ORIGINAL unit order=[19, 17, 13, 11, 5, 1, 20, 16, 10, 8, 4, 2]

After

SOURCE output sorts the mapped list numerically=[1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20];full-list equality=Yes;not a claim each point is fixed

Conditions: none

The source's second field is the sorted complete image list,not images in original unit order.Equal lists certify a permutation,not that multiplication by a fixes every point.The raw images are listed separately here.

Step identifiers and dependenciesRule: euler_sorted_mapped_units_not_original_image_order_or_identity · Uses: G001, S095









## Reverse checks


- S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S094 · premise_and_result_audit — Recompute Euler unit representative20 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S093 · premise_and_result_audit — Recompute Euler unit representative20 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S092 · premise_and_result_audit — Recompute Euler unit representative20 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S091 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative20 with exact rational arithmetic, not rounded decimals.
- S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S089 · premise_and_result_audit — Recompute Euler unit representative19 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S088 · premise_and_result_audit — Recompute Euler unit representative19 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S087 · premise_and_result_audit — Recompute Euler unit representative19 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S086 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative19 with exact rational arithmetic, not rounded decimals.
- S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S084 · premise_and_result_audit — Recompute Euler unit representative17 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S083 · premise_and_result_audit — Recompute Euler unit representative17 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S082 · premise_and_result_audit — Recompute Euler unit representative17 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S081 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative17 with exact rational arithmetic, not rounded decimals.
- S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S079 · premise_and_result_audit — Recompute Euler unit representative16 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S078 · premise_and_result_audit — Recompute Euler unit representative16 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S077 · premise_and_result_audit — Recompute Euler unit representative16 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S076 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative16 with exact rational arithmetic, not rounded decimals.
- S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S074 · premise_and_result_audit — Recompute Euler unit representative13 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S073 · premise_and_result_audit — Recompute Euler unit representative13 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S072 · premise_and_result_audit — Recompute Euler unit representative13 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S071 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative13 with exact rational arithmetic, not rounded decimals.
- S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S069 · premise_and_result_audit — Recompute Euler unit representative11 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S068 · premise_and_result_audit — Recompute Euler unit representative11 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S067 · premise_and_result_audit — Recompute Euler unit representative11 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S066 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative11 with exact rational arithmetic, not rounded decimals.
- S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S064 · premise_and_result_audit — Recompute Euler unit representative10 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S063 · premise_and_result_audit — Recompute Euler unit representative10 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S062 · premise_and_result_audit — Recompute Euler unit representative10 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S061 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative10 with exact rational arithmetic, not rounded decimals.
- S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S059 · premise_and_result_audit — Recompute Euler unit representative8 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S058 · premise_and_result_audit — Recompute Euler unit representative8 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S057 · premise_and_result_audit — Recompute Euler unit representative8 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S056 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative8 with exact rational arithmetic, not rounded decimals.
- 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 unit representative5 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S053 · premise_and_result_audit — Recompute Euler unit representative5 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S052 · premise_and_result_audit — Recompute Euler unit representative5 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S051 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative5 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 unit representative4 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S048 · premise_and_result_audit — Recompute Euler unit representative4 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S047 · premise_and_result_audit — Recompute Euler unit representative4 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S046 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative4 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 — Recompute Euler unit representative2 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S043 · premise_and_result_audit — Recompute Euler unit representative2 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S042 · premise_and_result_audit — Recompute Euler unit representative2 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S041 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative2 with exact rational arithmetic, not rounded decimals.
- S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S039 · premise_and_result_audit — Recompute Euler unit representative1 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S038 · premise_and_result_audit — Recompute Euler unit representative1 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S037 · premise_and_result_audit — Recompute Euler unit representative1 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S036 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative1 with exact rational arithmetic, not rounded decimals.
- S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — Recompute Euler ACTUAL inverse check normalized remainder with exact rational arithmetic, not rounded decimals.
- S032 · premise_and_result_audit — Recompute Euler ACTUAL inverse check floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S031 · premise_and_result_audit — Recompute Euler ACTUAL inverse check exact quotient with exact rational arithmetic, not rounded decimals.
- S030 · premise_and_result_audit — Recompute Euler ACTUAL base-unit inverse product with exact rational arithmetic, not rounded decimals.
- S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S028 · premise_and_result_audit — Recompute Euler complete reduced residue count phi with exact rational arithmetic, not rounded decimals.
- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

([1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], Yes, multiplication_by_a_permutes_reduced_residues)

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: 7f72d2eb2ca86b0c089b53f2d4e48d2f996bc36404e5ebe9992bf2c4b89ab6ac
- candidate record SHA-256: 72f858850ad7b91fcd8b64d43f774dcb70b5a460e552cd61efd65efcdc162a1d
- 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-0042.json)