# Question 20100: 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.

R0039-20100 · q-20261014-advance-20261107-r06-euler-theorem-number-theory-0050



## Original question and givens

Use integer modulus n=29≥2 and original integer base a=26. 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=29≥2 and original integer base a=26. 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(26, 29)

After

Euler ORIGINAL signed base normalization exact quotient = 26/29

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, 29)

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(26, 0)

After

Euler ORIGINAL signed base normalization normalized remainder = 26

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=26;modulus=29;exact ratio=26/29;floor quotient=0

After

value=q*n+remainder;normalized residue=26 in[0,29);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=[26, 29];nonnegative initial pair=[26, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[26, 29, 0, 26], [29, 26, 1, 3], [26, 3, 8, 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, S005





S007 · impliesBefore

Euler residue representative0:ORIGINAL gcd inputs=[0, 29];nonnegative initial pair=[0, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[0, 29, 0, 0]];terminal gcd=29

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, 29];nonnegative initial pair=[1, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[1, 29, 0, 1], [29, 1, 29, 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, 29];nonnegative initial pair=[2, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[2, 29, 0, 2], [29, 2, 14, 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, 29];nonnegative initial pair=[3, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[3, 29, 0, 3], [29, 3, 9, 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, S009





S011 · impliesBefore

Euler residue representative4:ORIGINAL gcd inputs=[4, 29];nonnegative initial pair=[4, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[4, 29, 0, 4], [29, 4, 7, 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, 29];nonnegative initial pair=[5, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[5, 29, 0, 5], [29, 5, 5, 4], [5, 4, 1, 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, S011





S013 · impliesBefore

Euler residue representative6:ORIGINAL gcd inputs=[6, 29];nonnegative initial pair=[6, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[6, 29, 0, 6], [29, 6, 4, 5], [6, 5, 1, 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, S012





S014 · impliesBefore

Euler residue representative7:ORIGINAL gcd inputs=[7, 29];nonnegative initial pair=[7, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[7, 29, 0, 7], [29, 7, 4, 1], [7, 1, 7, 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, 29];nonnegative initial pair=[8, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[8, 29, 0, 8], [29, 8, 3, 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, 29];nonnegative initial pair=[9, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[9, 29, 0, 9], [29, 9, 3, 2], [9, 2, 4, 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, S015





S017 · impliesBefore

Euler residue representative10:ORIGINAL gcd inputs=[10, 29];nonnegative initial pair=[10, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[10, 29, 0, 10], [29, 10, 2, 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, S016





S018 · impliesBefore

Euler residue representative11:ORIGINAL gcd inputs=[11, 29];nonnegative initial pair=[11, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[11, 29, 0, 11], [29, 11, 2, 7], [11, 7, 1, 4], [7, 4, 1, 3], [4, 3, 1, 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, S017





S019 · impliesBefore

Euler residue representative12:ORIGINAL gcd inputs=[12, 29];nonnegative initial pair=[12, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[12, 29, 0, 12], [29, 12, 2, 5], [12, 5, 2, 2], [5, 2, 2, 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, S018





S020 · impliesBefore

Euler residue representative13:ORIGINAL gcd inputs=[13, 29];nonnegative initial pair=[13, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[13, 29, 0, 13], [29, 13, 2, 3], [13, 3, 4, 1], [3, 1, 3, 0]];terminal gcd=1

Conditions: none

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

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





S021 · impliesBefore

Euler residue representative14:ORIGINAL gcd inputs=[14, 29];nonnegative initial pair=[14, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[14, 29, 0, 14], [29, 14, 2, 1], [14, 1, 14, 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, S020





S022 · impliesBefore

Euler residue representative15:ORIGINAL gcd inputs=[15, 29];nonnegative initial pair=[15, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[15, 29, 0, 15], [29, 15, 1, 14], [15, 14, 1, 1], [14, 1, 14, 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, S021





S023 · impliesBefore

Euler residue representative16:ORIGINAL gcd inputs=[16, 29];nonnegative initial pair=[16, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[16, 29, 0, 16], [29, 16, 1, 13], [16, 13, 1, 3], [13, 3, 4, 1], [3, 1, 3, 0]];terminal gcd=1

Conditions: none

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

Step identifiers and dependenciesRule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S022





S024 · impliesBefore

Euler residue representative17:ORIGINAL gcd inputs=[17, 29];nonnegative initial pair=[17, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[17, 29, 0, 17], [29, 17, 1, 12], [17, 12, 1, 5], [12, 5, 2, 2], [5, 2, 2, 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, S023





S025 · impliesBefore

Euler residue representative18:ORIGINAL gcd inputs=[18, 29];nonnegative initial pair=[18, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[18, 29, 0, 18], [29, 18, 1, 11], [18, 11, 1, 7], [11, 7, 1, 4], [7, 4, 1, 3], [4, 3, 1, 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, S024





S026 · impliesBefore

Euler residue representative19:ORIGINAL gcd inputs=[19, 29];nonnegative initial pair=[19, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[19, 29, 0, 19], [29, 19, 1, 10], [19, 10, 1, 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, S025





S027 · impliesBefore

Euler residue representative20:ORIGINAL gcd inputs=[20, 29];nonnegative initial pair=[20, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[20, 29, 0, 20], [29, 20, 1, 9], [20, 9, 2, 2], [9, 2, 4, 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, S026





S028 · impliesBefore

Euler residue representative21:ORIGINAL gcd inputs=[21, 29];nonnegative initial pair=[21, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[21, 29, 0, 21], [29, 21, 1, 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, S027





S029 · impliesBefore

Euler residue representative22:ORIGINAL gcd inputs=[22, 29];nonnegative initial pair=[22, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[22, 29, 0, 22], [29, 22, 1, 7], [22, 7, 3, 1], [7, 1, 7, 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, S028





S030 · impliesBefore

Euler residue representative23:ORIGINAL gcd inputs=[23, 29];nonnegative initial pair=[23, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[23, 29, 0, 23], [29, 23, 1, 6], [23, 6, 3, 5], [6, 5, 1, 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, S029





S031 · impliesBefore

Euler residue representative24:ORIGINAL gcd inputs=[24, 29];nonnegative initial pair=[24, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[24, 29, 0, 24], [29, 24, 1, 5], [24, 5, 4, 4], [5, 4, 1, 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, S030





S032 · impliesBefore

Euler residue representative25:ORIGINAL gcd inputs=[25, 29];nonnegative initial pair=[25, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[25, 29, 0, 25], [29, 25, 1, 4], [25, 4, 6, 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, S031





S033 · impliesBefore

Euler residue representative26:ORIGINAL gcd inputs=[26, 29];nonnegative initial pair=[26, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[26, 29, 0, 26], [29, 26, 1, 3], [26, 3, 8, 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, S032





S034 · impliesBefore

Euler residue representative27:ORIGINAL gcd inputs=[27, 29];nonnegative initial pair=[27, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[27, 29, 0, 27], [29, 27, 1, 2], [27, 2, 13, 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, S033





S035 · impliesBefore

Euler residue representative28:ORIGINAL gcd inputs=[28, 29];nonnegative initial pair=[28, 29]

After

ALL exact Euclidean rows[left,right,q,r]=[[28, 29, 0, 28], [29, 28, 1, 1], [28, 1, 28, 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, S034





S036 · calculationBefore

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

After

Euler complete reduced residue count phi = 28

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 · impliesBefore

COMPLETE original residue carrier0..28;ALL [representative,gcd,unit] rows=[[0, 29, No], [1, 1, Yes], [2, 1, Yes], [3, 1, Yes], [4, 1, Yes], [5, 1, Yes], [6, 1, Yes], [7, 1, Yes], [8, 1, Yes], [9, 1, Yes], [10, 1, Yes], [11, 1, Yes], [12, 1, Yes], [13, 1, Yes], [14, 1, Yes], [15, 1, Yes], [16, 1, Yes], [17, 1, Yes], [18, 1, Yes], [19, 1, Yes], [20, 1, Yes], [21, 1, Yes], [22, 1, Yes], [23, 1, Yes], [24, 1, Yes], [25, 1, Yes], [26, 1, Yes], [27, 1, Yes], [28, 1, Yes]]

After

increasing COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28];phi(n)=28;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, S036





S038 · calculationBefore

Euler ACTUAL base-unit inverse product := mul(26, 19)

After

Euler ACTUAL base-unit inverse product = 494

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 ACTUAL inverse check exact quotient := div(494, 29)

After

Euler ACTUAL inverse check exact quotient = 494/29

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 ACTUAL inverse check floor quotient multiple := mul(17, 29)

After

Euler ACTUAL inverse check floor quotient multiple = 493

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 · calculationBefore

Euler ACTUAL inverse check normalized remainder := sub(494, 493)

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, S040





S042 · impliesBefore

Euler ACTUAL inverse check:ORIGINAL signed integer=494;modulus=29;exact ratio=494/29;floor quotient=17

After

value=q*n+remainder;normalized residue=1 in[0,29);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, S041





S043 · impliesBefore

ACTUAL modulus n=29;gcd(original a=26,n)=1;normalized base=26;ACTUAL inverse=19;COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]

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, S042





S044 · calculationBefore

Euler ORIGINAL a times unit representative1 := mul(26, 1)

After

Euler ORIGINAL a times unit representative1 = 26

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 unit representative1 raw image exact quotient := div(26, 29)

After

Euler unit representative1 raw image exact quotient = 26/29

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 unit representative1 raw image floor quotient multiple := mul(0, 29)

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, S045





S047 · calculationBefore

Euler unit representative1 raw image normalized remainder := sub(26, 0)

After

Euler unit representative1 raw image normalized remainder = 26

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 unit representative1 raw image:ORIGINAL signed integer=26;modulus=29;exact ratio=26/29;floor quotient=0

After

value=q*n+remainder;normalized residue=26 in[0,29);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 · calculationBefore

Euler ORIGINAL a times unit representative2 := mul(26, 2)

After

Euler ORIGINAL a times unit representative2 = 52

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 · calculationBefore

Euler unit representative2 raw image exact quotient := div(52, 29)

After

Euler unit representative2 raw image exact quotient = 52/29

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, S049





S051 · calculationBefore

Euler unit representative2 raw image floor quotient multiple := mul(1, 29)

After

Euler unit representative2 raw image floor quotient multiple = 29

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 representative2 raw image normalized remainder := sub(52, 29)

After

Euler unit representative2 raw image normalized remainder = 23

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 · impliesBefore

Euler unit representative2 raw image:ORIGINAL signed integer=52;modulus=29;exact ratio=52/29;floor quotient=1

After

value=q*n+remainder;normalized residue=23 in[0,29);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, S052





S054 · calculationBefore

Euler ORIGINAL a times unit representative3 := mul(26, 3)

After

Euler ORIGINAL a times unit representative3 = 78

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 · calculationBefore

Euler unit representative3 raw image exact quotient := div(78, 29)

After

Euler unit representative3 raw image exact quotient = 78/29

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, S054





S056 · calculationBefore

Euler unit representative3 raw image floor quotient multiple := mul(2, 29)

After

Euler unit representative3 raw image floor quotient multiple = 58

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 representative3 raw image normalized remainder := sub(78, 58)

After

Euler unit representative3 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, S056





S058 · impliesBefore

Euler unit representative3 raw image:ORIGINAL signed integer=78;modulus=29;exact ratio=78/29;floor quotient=2

After

value=q*n+remainder;normalized residue=20 in[0,29);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, S057





S059 · calculationBefore

Euler ORIGINAL a times unit representative4 := mul(26, 4)

After

Euler ORIGINAL a times unit representative4 = 104

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 · calculationBefore

Euler unit representative4 raw image exact quotient := div(104, 29)

After

Euler unit representative4 raw image exact quotient = 104/29

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, S059





S061 · calculationBefore

Euler unit representative4 raw image floor quotient multiple := mul(3, 29)

After

Euler unit representative4 raw image floor quotient multiple = 87

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 representative4 raw image normalized remainder := sub(104, 87)

After

Euler unit representative4 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, S061





S063 · impliesBefore

Euler unit representative4 raw image:ORIGINAL signed integer=104;modulus=29;exact ratio=104/29;floor quotient=3

After

value=q*n+remainder;normalized residue=17 in[0,29);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, S062





S064 · calculationBefore

Euler ORIGINAL a times unit representative5 := mul(26, 5)

After

Euler ORIGINAL a times unit representative5 = 130

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 · calculationBefore

Euler unit representative5 raw image exact quotient := div(130, 29)

After

Euler unit representative5 raw image exact quotient = 130/29

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, S064





S066 · calculationBefore

Euler unit representative5 raw image floor quotient multiple := mul(4, 29)

After

Euler unit representative5 raw image floor quotient multiple = 116

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 representative5 raw image normalized remainder := sub(130, 116)

After

Euler unit representative5 raw image normalized remainder = 14

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 · impliesBefore

Euler unit representative5 raw image:ORIGINAL signed integer=130;modulus=29;exact ratio=130/29;floor quotient=4

After

value=q*n+remainder;normalized residue=14 in[0,29);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, S067





S069 · calculationBefore

Euler ORIGINAL a times unit representative6 := mul(26, 6)

After

Euler ORIGINAL a times unit representative6 = 156

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 · calculationBefore

Euler unit representative6 raw image exact quotient := div(156, 29)

After

Euler unit representative6 raw image exact quotient = 156/29

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, S069





S071 · calculationBefore

Euler unit representative6 raw image floor quotient multiple := mul(5, 29)

After

Euler unit representative6 raw image floor quotient multiple = 145

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 representative6 raw image normalized remainder := sub(156, 145)

After

Euler unit representative6 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, S071





S073 · impliesBefore

Euler unit representative6 raw image:ORIGINAL signed integer=156;modulus=29;exact ratio=156/29;floor quotient=5

After

value=q*n+remainder;normalized residue=11 in[0,29);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, S072





S074 · calculationBefore

Euler ORIGINAL a times unit representative7 := mul(26, 7)

After

Euler ORIGINAL a times unit representative7 = 182

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 · calculationBefore

Euler unit representative7 raw image exact quotient := div(182, 29)

After

Euler unit representative7 raw image exact quotient = 182/29

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, S074





S076 · calculationBefore

Euler unit representative7 raw image floor quotient multiple := mul(6, 29)

After

Euler unit representative7 raw image floor quotient multiple = 174

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 representative7 raw image normalized remainder := sub(182, 174)

After

Euler unit representative7 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, S076





S078 · impliesBefore

Euler unit representative7 raw image:ORIGINAL signed integer=182;modulus=29;exact ratio=182/29;floor quotient=6

After

value=q*n+remainder;normalized residue=8 in[0,29);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, S077





S079 · calculationBefore

Euler ORIGINAL a times unit representative8 := mul(26, 8)

After

Euler ORIGINAL a times unit representative8 = 208

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 · calculationBefore

Euler unit representative8 raw image exact quotient := div(208, 29)

After

Euler unit representative8 raw image exact quotient = 208/29

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, S079





S081 · calculationBefore

Euler unit representative8 raw image floor quotient multiple := mul(7, 29)

After

Euler unit representative8 raw image floor quotient multiple = 203

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 representative8 raw image normalized remainder := sub(208, 203)

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, S081





S083 · impliesBefore

Euler unit representative8 raw image:ORIGINAL signed integer=208;modulus=29;exact ratio=208/29;floor quotient=7

After

value=q*n+remainder;normalized residue=5 in[0,29);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, S082





S084 · calculationBefore

Euler ORIGINAL a times unit representative9 := mul(26, 9)

After

Euler ORIGINAL a times unit representative9 = 234

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 · calculationBefore

Euler unit representative9 raw image exact quotient := div(234, 29)

After

Euler unit representative9 raw image exact quotient = 234/29

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, S084





S086 · calculationBefore

Euler unit representative9 raw image floor quotient multiple := mul(8, 29)

After

Euler unit representative9 raw image floor quotient multiple = 232

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 representative9 raw image normalized remainder := sub(234, 232)

After

Euler unit representative9 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, S086





S088 · impliesBefore

Euler unit representative9 raw image:ORIGINAL signed integer=234;modulus=29;exact ratio=234/29;floor quotient=8

After

value=q*n+remainder;normalized residue=2 in[0,29);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, S087





S089 · calculationBefore

Euler ORIGINAL a times unit representative10 := mul(26, 10)

After

Euler ORIGINAL a times unit representative10 = 260

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 · calculationBefore

Euler unit representative10 raw image exact quotient := div(260, 29)

After

Euler unit representative10 raw image exact quotient = 260/29

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, S089





S091 · calculationBefore

Euler unit representative10 raw image floor quotient multiple := mul(8, 29)

After

Euler unit representative10 raw image floor quotient multiple = 232

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 representative10 raw image normalized remainder := sub(260, 232)

After

Euler unit representative10 raw image normalized remainder = 28

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 · impliesBefore

Euler unit representative10 raw image:ORIGINAL signed integer=260;modulus=29;exact ratio=260/29;floor quotient=8

After

value=q*n+remainder;normalized residue=28 in[0,29);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, S092





S094 · calculationBefore

Euler ORIGINAL a times unit representative11 := mul(26, 11)

After

Euler ORIGINAL a times unit representative11 = 286

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 · calculationBefore

Euler unit representative11 raw image exact quotient := div(286, 29)

After

Euler unit representative11 raw image exact quotient = 286/29

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, S094





S096 · calculationBefore

Euler unit representative11 raw image floor quotient multiple := mul(9, 29)

After

Euler unit representative11 raw image floor quotient multiple = 261

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, S095





S097 · calculationBefore

Euler unit representative11 raw image normalized remainder := sub(286, 261)

After

Euler unit representative11 raw image normalized remainder = 25

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, S096





S098 · impliesBefore

Euler unit representative11 raw image:ORIGINAL signed integer=286;modulus=29;exact ratio=286/29;floor quotient=9

After

value=q*n+remainder;normalized residue=25 in[0,29);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, S097





S099 · calculationBefore

Euler ORIGINAL a times unit representative12 := mul(26, 12)

After

Euler ORIGINAL a times unit representative12 = 312

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, S098





S100 · calculationBefore

Euler unit representative12 raw image exact quotient := div(312, 29)

After

Euler unit representative12 raw image exact quotient = 312/29

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, S099





S101 · calculationBefore

Euler unit representative12 raw image floor quotient multiple := mul(10, 29)

After

Euler unit representative12 raw image floor quotient multiple = 290

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, S100





S102 · calculationBefore

Euler unit representative12 raw image normalized remainder := sub(312, 290)

After

Euler unit representative12 raw image normalized remainder = 22

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, S101





S103 · impliesBefore

Euler unit representative12 raw image:ORIGINAL signed integer=312;modulus=29;exact ratio=312/29;floor quotient=10

After

value=q*n+remainder;normalized residue=22 in[0,29);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, S102





S104 · calculationBefore

Euler ORIGINAL a times unit representative13 := mul(26, 13)

After

Euler ORIGINAL a times unit representative13 = 338

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, S103





S105 · calculationBefore

Euler unit representative13 raw image exact quotient := div(338, 29)

After

Euler unit representative13 raw image exact quotient = 338/29

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, S104





S106 · calculationBefore

Euler unit representative13 raw image floor quotient multiple := mul(11, 29)

After

Euler unit representative13 raw image floor quotient multiple = 319

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, S105





S107 · calculationBefore

Euler unit representative13 raw image normalized remainder := sub(338, 319)

After

Euler unit representative13 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, S106





S108 · impliesBefore

Euler unit representative13 raw image:ORIGINAL signed integer=338;modulus=29;exact ratio=338/29;floor quotient=11

After

value=q*n+remainder;normalized residue=19 in[0,29);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, S107





S109 · calculationBefore

Euler ORIGINAL a times unit representative14 := mul(26, 14)

After

Euler ORIGINAL a times unit representative14 = 364

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, S108





S110 · calculationBefore

Euler unit representative14 raw image exact quotient := div(364, 29)

After

Euler unit representative14 raw image exact quotient = 364/29

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, S109





S111 · calculationBefore

Euler unit representative14 raw image floor quotient multiple := mul(12, 29)

After

Euler unit representative14 raw image floor quotient multiple = 348

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, S110





S112 · calculationBefore

Euler unit representative14 raw image normalized remainder := sub(364, 348)

After

Euler unit representative14 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, S111





S113 · impliesBefore

Euler unit representative14 raw image:ORIGINAL signed integer=364;modulus=29;exact ratio=364/29;floor quotient=12

After

value=q*n+remainder;normalized residue=16 in[0,29);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, S112





S114 · calculationBefore

Euler ORIGINAL a times unit representative15 := mul(26, 15)

After

Euler ORIGINAL a times unit representative15 = 390

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, S113





S115 · calculationBefore

Euler unit representative15 raw image exact quotient := div(390, 29)

After

Euler unit representative15 raw image exact quotient = 390/29

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, S114





S116 · calculationBefore

Euler unit representative15 raw image floor quotient multiple := mul(13, 29)

After

Euler unit representative15 raw image floor quotient multiple = 377

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, S115





S117 · calculationBefore

Euler unit representative15 raw image normalized remainder := sub(390, 377)

After

Euler unit representative15 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, S116





S118 · impliesBefore

Euler unit representative15 raw image:ORIGINAL signed integer=390;modulus=29;exact ratio=390/29;floor quotient=13

After

value=q*n+remainder;normalized residue=13 in[0,29);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, S117





S119 · calculationBefore

Euler ORIGINAL a times unit representative16 := mul(26, 16)

After

Euler ORIGINAL a times unit representative16 = 416

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, S118





S120 · calculationBefore

Euler unit representative16 raw image exact quotient := div(416, 29)

After

Euler unit representative16 raw image exact quotient = 416/29

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, S119





S121 · calculationBefore

Euler unit representative16 raw image floor quotient multiple := mul(14, 29)

After

Euler unit representative16 raw image floor quotient multiple = 406

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, S120





S122 · calculationBefore

Euler unit representative16 raw image normalized remainder := sub(416, 406)

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, S121





S123 · impliesBefore

Euler unit representative16 raw image:ORIGINAL signed integer=416;modulus=29;exact ratio=416/29;floor quotient=14

After

value=q*n+remainder;normalized residue=10 in[0,29);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, S122





S124 · calculationBefore

Euler ORIGINAL a times unit representative17 := mul(26, 17)

After

Euler ORIGINAL a times unit representative17 = 442

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, S123





S125 · calculationBefore

Euler unit representative17 raw image exact quotient := div(442, 29)

After

Euler unit representative17 raw image exact quotient = 442/29

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, S124





S126 · calculationBefore

Euler unit representative17 raw image floor quotient multiple := mul(15, 29)

After

Euler unit representative17 raw image floor quotient multiple = 435

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, S125





S127 · calculationBefore

Euler unit representative17 raw image normalized remainder := sub(442, 435)

After

Euler unit representative17 raw image 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, S126





S128 · impliesBefore

Euler unit representative17 raw image:ORIGINAL signed integer=442;modulus=29;exact ratio=442/29;floor quotient=15

After

value=q*n+remainder;normalized residue=7 in[0,29);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, S127





S129 · calculationBefore

Euler ORIGINAL a times unit representative18 := mul(26, 18)

After

Euler ORIGINAL a times unit representative18 = 468

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, S128





S130 · calculationBefore

Euler unit representative18 raw image exact quotient := div(468, 29)

After

Euler unit representative18 raw image exact quotient = 468/29

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, S129





S131 · calculationBefore

Euler unit representative18 raw image floor quotient multiple := mul(16, 29)

After

Euler unit representative18 raw image floor quotient multiple = 464

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, S130





S132 · calculationBefore

Euler unit representative18 raw image normalized remainder := sub(468, 464)

After

Euler unit representative18 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, S131





S133 · impliesBefore

Euler unit representative18 raw image:ORIGINAL signed integer=468;modulus=29;exact ratio=468/29;floor quotient=16

After

value=q*n+remainder;normalized residue=4 in[0,29);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, S132





S134 · calculationBefore

Euler ORIGINAL a times unit representative19 := mul(26, 19)

After

Euler ORIGINAL a times unit representative19 = 494

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, S133





S135 · calculationBefore

Euler unit representative19 raw image exact quotient := div(494, 29)

After

Euler unit representative19 raw image exact quotient = 494/29

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, S134





S136 · calculationBefore

Euler unit representative19 raw image floor quotient multiple := mul(17, 29)

After

Euler unit representative19 raw image floor quotient multiple = 493

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, S135





S137 · calculationBefore

Euler unit representative19 raw image normalized remainder := sub(494, 493)

After

Euler unit representative19 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, S136





S138 · impliesBefore

Euler unit representative19 raw image:ORIGINAL signed integer=494;modulus=29;exact ratio=494/29;floor quotient=17

After

value=q*n+remainder;normalized residue=1 in[0,29);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, S137





S139 · calculationBefore

Euler ORIGINAL a times unit representative20 := mul(26, 20)

After

Euler ORIGINAL a times unit representative20 = 520

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, S138





S140 · calculationBefore

Euler unit representative20 raw image exact quotient := div(520, 29)

After

Euler unit representative20 raw image exact quotient = 520/29

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, S139





S141 · calculationBefore

Euler unit representative20 raw image floor quotient multiple := mul(17, 29)

After

Euler unit representative20 raw image floor quotient multiple = 493

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, S140





S142 · calculationBefore

Euler unit representative20 raw image normalized remainder := sub(520, 493)

After

Euler unit representative20 raw image normalized remainder = 27

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, S141





S143 · impliesBefore

Euler unit representative20 raw image:ORIGINAL signed integer=520;modulus=29;exact ratio=520/29;floor quotient=17

After

value=q*n+remainder;normalized residue=27 in[0,29);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, S142





S144 · calculationBefore

Euler ORIGINAL a times unit representative21 := mul(26, 21)

After

Euler ORIGINAL a times unit representative21 = 546

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, S143





S145 · calculationBefore

Euler unit representative21 raw image exact quotient := div(546, 29)

After

Euler unit representative21 raw image exact quotient = 546/29

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, S144





S146 · calculationBefore

Euler unit representative21 raw image floor quotient multiple := mul(18, 29)

After

Euler unit representative21 raw image floor quotient multiple = 522

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, S145





S147 · calculationBefore

Euler unit representative21 raw image normalized remainder := sub(546, 522)

After

Euler unit representative21 raw image normalized remainder = 24

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, S146





S148 · impliesBefore

Euler unit representative21 raw image:ORIGINAL signed integer=546;modulus=29;exact ratio=546/29;floor quotient=18

After

value=q*n+remainder;normalized residue=24 in[0,29);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, S147





S149 · calculationBefore

Euler ORIGINAL a times unit representative22 := mul(26, 22)

After

Euler ORIGINAL a times unit representative22 = 572

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, S148





S150 · calculationBefore

Euler unit representative22 raw image exact quotient := div(572, 29)

After

Euler unit representative22 raw image exact quotient = 572/29

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, S149





S151 · calculationBefore

Euler unit representative22 raw image floor quotient multiple := mul(19, 29)

After

Euler unit representative22 raw image floor quotient multiple = 551

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, S150





S152 · calculationBefore

Euler unit representative22 raw image normalized remainder := sub(572, 551)

After

Euler unit representative22 raw image normalized remainder = 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, S151





S153 · impliesBefore

Euler unit representative22 raw image:ORIGINAL signed integer=572;modulus=29;exact ratio=572/29;floor quotient=19

After

value=q*n+remainder;normalized residue=21 in[0,29);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, S152





S154 · calculationBefore

Euler ORIGINAL a times unit representative23 := mul(26, 23)

After

Euler ORIGINAL a times unit representative23 = 598

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, S153





S155 · calculationBefore

Euler unit representative23 raw image exact quotient := div(598, 29)

After

Euler unit representative23 raw image exact quotient = 598/29

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, S154





S156 · calculationBefore

Euler unit representative23 raw image floor quotient multiple := mul(20, 29)

After

Euler unit representative23 raw image floor quotient multiple = 580

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, S155





S157 · calculationBefore

Euler unit representative23 raw image normalized remainder := sub(598, 580)

After

Euler unit representative23 raw image normalized remainder = 18

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, S156





S158 · impliesBefore

Euler unit representative23 raw image:ORIGINAL signed integer=598;modulus=29;exact ratio=598/29;floor quotient=20

After

value=q*n+remainder;normalized residue=18 in[0,29);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, S157





S159 · calculationBefore

Euler ORIGINAL a times unit representative24 := mul(26, 24)

After

Euler ORIGINAL a times unit representative24 = 624

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, S158





S160 · calculationBefore

Euler unit representative24 raw image exact quotient := div(624, 29)

After

Euler unit representative24 raw image exact quotient = 624/29

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, S159





S161 · calculationBefore

Euler unit representative24 raw image floor quotient multiple := mul(21, 29)

After

Euler unit representative24 raw image floor quotient multiple = 609

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, S160





S162 · calculationBefore

Euler unit representative24 raw image normalized remainder := sub(624, 609)

After

Euler unit representative24 raw image normalized remainder = 15

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, S161





S163 · impliesBefore

Euler unit representative24 raw image:ORIGINAL signed integer=624;modulus=29;exact ratio=624/29;floor quotient=21

After

value=q*n+remainder;normalized residue=15 in[0,29);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, S162





S164 · calculationBefore

Euler ORIGINAL a times unit representative25 := mul(26, 25)

After

Euler ORIGINAL a times unit representative25 = 650

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, S163





S165 · calculationBefore

Euler unit representative25 raw image exact quotient := div(650, 29)

After

Euler unit representative25 raw image exact quotient = 650/29

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, S164





S166 · calculationBefore

Euler unit representative25 raw image floor quotient multiple := mul(22, 29)

After

Euler unit representative25 raw image floor quotient multiple = 638

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, S165





S167 · calculationBefore

Euler unit representative25 raw image normalized remainder := sub(650, 638)

After

Euler unit representative25 raw image normalized remainder = 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, S166





S168 · impliesBefore

Euler unit representative25 raw image:ORIGINAL signed integer=650;modulus=29;exact ratio=650/29;floor quotient=22

After

value=q*n+remainder;normalized residue=12 in[0,29);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, S167





S169 · calculationBefore

Euler ORIGINAL a times unit representative26 := mul(26, 26)

After

Euler ORIGINAL a times unit representative26 = 676

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, S168





S170 · calculationBefore

Euler unit representative26 raw image exact quotient := div(676, 29)

After

Euler unit representative26 raw image exact quotient = 676/29

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, S169





S171 · calculationBefore

Euler unit representative26 raw image floor quotient multiple := mul(23, 29)

After

Euler unit representative26 raw image floor quotient multiple = 667

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, S170





S172 · calculationBefore

Euler unit representative26 raw image normalized remainder := sub(676, 667)

After

Euler unit representative26 raw image 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, S171





S173 · impliesBefore

Euler unit representative26 raw image:ORIGINAL signed integer=676;modulus=29;exact ratio=676/29;floor quotient=23

After

value=q*n+remainder;normalized residue=9 in[0,29);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, S172





S174 · calculationBefore

Euler ORIGINAL a times unit representative27 := mul(26, 27)

After

Euler ORIGINAL a times unit representative27 = 702

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, S173





S175 · calculationBefore

Euler unit representative27 raw image exact quotient := div(702, 29)

After

Euler unit representative27 raw image exact quotient = 702/29

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, S174





S176 · calculationBefore

Euler unit representative27 raw image floor quotient multiple := mul(24, 29)

After

Euler unit representative27 raw image floor quotient multiple = 696

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, S175





S177 · calculationBefore

Euler unit representative27 raw image normalized remainder := sub(702, 696)

After

Euler unit representative27 raw image normalized remainder = 6

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, S176





S178 · impliesBefore

Euler unit representative27 raw image:ORIGINAL signed integer=702;modulus=29;exact ratio=702/29;floor quotient=24

After

value=q*n+remainder;normalized residue=6 in[0,29);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, S177





S179 · calculationBefore

Euler ORIGINAL a times unit representative28 := mul(26, 28)

After

Euler ORIGINAL a times unit representative28 = 728

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, S178





S180 · calculationBefore

Euler unit representative28 raw image exact quotient := div(728, 29)

After

Euler unit representative28 raw image exact quotient = 728/29

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, S179





S181 · calculationBefore

Euler unit representative28 raw image floor quotient multiple := mul(25, 29)

After

Euler unit representative28 raw image floor quotient multiple = 725

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, S180





S182 · calculationBefore

Euler unit representative28 raw image normalized remainder := sub(728, 725)

After

Euler unit representative28 raw image normalized remainder = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S181





S183 · impliesBefore

Euler unit representative28 raw image:ORIGINAL signed integer=728;modulus=29;exact ratio=728/29;floor quotient=25

After

value=q*n+remainder;normalized residue=3 in[0,29);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, S182





S184 · impliesBefore

COMPLETE original increasing U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28];actual images in ORIGINAL unit order=[26, 23, 20, 17, 14, 11, 8, 5, 2, 28, 25, 22, 19, 16, 13, 10, 7, 4, 1, 27, 24, 21, 18, 15, 12, 9, 6, 3]

After

SOURCE output sorts the mapped list numerically=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28];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, S183









## Reverse checks


- S184 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S183 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S182 · premise_and_result_audit — Recompute Euler unit representative28 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S181 · premise_and_result_audit — Recompute Euler unit representative28 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S180 · premise_and_result_audit — Recompute Euler unit representative28 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S179 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative28 with exact rational arithmetic, not rounded decimals.
- S178 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S177 · premise_and_result_audit — Recompute Euler unit representative27 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S176 · premise_and_result_audit — Recompute Euler unit representative27 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S175 · premise_and_result_audit — Recompute Euler unit representative27 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S174 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative27 with exact rational arithmetic, not rounded decimals.
- S173 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S172 · premise_and_result_audit — Recompute Euler unit representative26 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S171 · premise_and_result_audit — Recompute Euler unit representative26 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S170 · premise_and_result_audit — Recompute Euler unit representative26 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S169 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative26 with exact rational arithmetic, not rounded decimals.
- S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S167 · premise_and_result_audit — Recompute Euler unit representative25 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S166 · premise_and_result_audit — Recompute Euler unit representative25 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S165 · premise_and_result_audit — Recompute Euler unit representative25 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S164 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative25 with exact rational arithmetic, not rounded decimals.
- S163 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S162 · premise_and_result_audit — Recompute Euler unit representative24 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S161 · premise_and_result_audit — Recompute Euler unit representative24 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S160 · premise_and_result_audit — Recompute Euler unit representative24 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S159 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative24 with exact rational arithmetic, not rounded decimals.
- S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S157 · premise_and_result_audit — Recompute Euler unit representative23 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S156 · premise_and_result_audit — Recompute Euler unit representative23 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S155 · premise_and_result_audit — Recompute Euler unit representative23 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S154 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative23 with exact rational arithmetic, not rounded decimals.
- S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S152 · premise_and_result_audit — Recompute Euler unit representative22 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S151 · premise_and_result_audit — Recompute Euler unit representative22 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S150 · premise_and_result_audit — Recompute Euler unit representative22 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S149 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative22 with exact rational arithmetic, not rounded decimals.
- S148 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S147 · premise_and_result_audit — Recompute Euler unit representative21 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S146 · premise_and_result_audit — Recompute Euler unit representative21 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S145 · premise_and_result_audit — Recompute Euler unit representative21 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S144 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative21 with exact rational arithmetic, not rounded decimals.
- S143 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S142 · premise_and_result_audit — Recompute Euler unit representative20 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S141 · premise_and_result_audit — Recompute Euler unit representative20 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S140 · premise_and_result_audit — Recompute Euler unit representative20 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S139 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative20 with exact rational arithmetic, not rounded decimals.
- S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S137 · premise_and_result_audit — Recompute Euler unit representative19 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S136 · premise_and_result_audit — Recompute Euler unit representative19 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S135 · premise_and_result_audit — Recompute Euler unit representative19 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S134 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative19 with exact rational arithmetic, not rounded decimals.
- S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S132 · premise_and_result_audit — Recompute Euler unit representative18 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S131 · premise_and_result_audit — Recompute Euler unit representative18 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S130 · premise_and_result_audit — Recompute Euler unit representative18 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S129 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative18 with exact rational arithmetic, not rounded decimals.
- S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S127 · premise_and_result_audit — Recompute Euler unit representative17 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S126 · premise_and_result_audit — Recompute Euler unit representative17 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S125 · premise_and_result_audit — Recompute Euler unit representative17 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S124 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative17 with exact rational arithmetic, not rounded decimals.
- S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S122 · premise_and_result_audit — Recompute Euler unit representative16 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S121 · premise_and_result_audit — Recompute Euler unit representative16 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S120 · premise_and_result_audit — Recompute Euler unit representative16 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S119 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative16 with exact rational arithmetic, not rounded decimals.
- S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S117 · premise_and_result_audit — Recompute Euler unit representative15 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S116 · premise_and_result_audit — Recompute Euler unit representative15 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S115 · premise_and_result_audit — Recompute Euler unit representative15 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S114 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative15 with exact rational arithmetic, not rounded decimals.
- S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S112 · premise_and_result_audit — Recompute Euler unit representative14 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S111 · premise_and_result_audit — Recompute Euler unit representative14 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S110 · premise_and_result_audit — Recompute Euler unit representative14 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S109 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative14 with exact rational arithmetic, not rounded decimals.
- S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S107 · premise_and_result_audit — Recompute Euler unit representative13 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S106 · premise_and_result_audit — Recompute Euler unit representative13 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S105 · premise_and_result_audit — Recompute Euler unit representative13 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S104 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative13 with exact rational arithmetic, not rounded decimals.
- S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S102 · premise_and_result_audit — Recompute Euler unit representative12 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S101 · premise_and_result_audit — Recompute Euler unit representative12 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S100 · premise_and_result_audit — Recompute Euler unit representative12 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S099 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative12 with exact rational arithmetic, not rounded decimals.
- S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S097 · premise_and_result_audit — Recompute Euler unit representative11 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S096 · premise_and_result_audit — Recompute Euler unit representative11 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S095 · premise_and_result_audit — Recompute Euler unit representative11 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S094 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative11 with exact rational arithmetic, not rounded decimals.
- S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S092 · premise_and_result_audit — Recompute Euler unit representative10 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S091 · premise_and_result_audit — Recompute Euler unit representative10 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S090 · premise_and_result_audit — Recompute Euler unit representative10 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S089 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative10 with exact rational arithmetic, not rounded decimals.
- S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S087 · premise_and_result_audit — Recompute Euler unit representative9 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S086 · premise_and_result_audit — Recompute Euler unit representative9 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S085 · premise_and_result_audit — Recompute Euler unit representative9 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S084 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative9 with exact rational arithmetic, not rounded decimals.
- S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S082 · premise_and_result_audit — Recompute Euler unit representative8 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S081 · premise_and_result_audit — Recompute Euler unit representative8 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S080 · premise_and_result_audit — Recompute Euler unit representative8 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S079 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative8 with exact rational arithmetic, not rounded decimals.
- S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S077 · premise_and_result_audit — Recompute Euler unit representative7 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S076 · premise_and_result_audit — Recompute Euler unit representative7 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S075 · premise_and_result_audit — Recompute Euler unit representative7 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S074 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative7 with exact rational arithmetic, not rounded decimals.
- S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S072 · premise_and_result_audit — Recompute Euler unit representative6 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S071 · premise_and_result_audit — Recompute Euler unit representative6 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S070 · premise_and_result_audit — Recompute Euler unit representative6 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S069 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative6 with exact rational arithmetic, not rounded decimals.
- S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S067 · premise_and_result_audit — Recompute Euler unit representative5 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S066 · premise_and_result_audit — Recompute Euler unit representative5 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S065 · premise_and_result_audit — Recompute Euler unit representative5 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S064 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative5 with exact rational arithmetic, not rounded decimals.
- S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S062 · premise_and_result_audit — Recompute Euler unit representative4 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S061 · premise_and_result_audit — Recompute Euler unit representative4 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S060 · premise_and_result_audit — Recompute Euler unit representative4 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S059 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative4 with exact rational arithmetic, not rounded decimals.
- S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S057 · premise_and_result_audit — Recompute Euler unit representative3 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S056 · premise_and_result_audit — Recompute Euler unit representative3 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S055 · premise_and_result_audit — Recompute Euler unit representative3 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S054 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative3 with exact rational arithmetic, not rounded decimals.
- S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S052 · premise_and_result_audit — Recompute Euler unit representative2 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S051 · premise_and_result_audit — Recompute Euler unit representative2 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S050 · premise_and_result_audit — Recompute Euler unit representative2 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S049 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative2 with exact rational arithmetic, not rounded decimals.
- 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 unit representative1 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
- S046 · premise_and_result_audit — Recompute Euler unit representative1 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S045 · premise_and_result_audit — Recompute Euler unit representative1 raw image exact quotient with exact rational arithmetic, not rounded decimals.
- S044 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative1 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 — Recompute Euler ACTUAL inverse check normalized remainder with exact rational arithmetic, not rounded decimals.
- S040 · premise_and_result_audit — Recompute Euler ACTUAL inverse check floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S039 · premise_and_result_audit — Recompute Euler ACTUAL inverse check exact quotient with exact rational arithmetic, not rounded decimals.
- S038 · premise_and_result_audit — Recompute Euler ACTUAL base-unit inverse product with exact rational arithmetic, not rounded decimals.
- S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S036 · premise_and_result_audit — Recompute Euler complete reduced residue count phi 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S026 · premise_and_result_audit — 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, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], 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: 8597de7a162277877a8de1a15604f3ece66a928137499b003ed42d9a0165072b
- candidate record SHA-256: b59a19b12db90f4bb44bbbf5553d86135b1ab362444b95d1336b102877d8be3e
- 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-0050.json)