Question 20069: A Nonunit Fails Euler Applicability, Not a Primality Test

Original question and givens

Use integer modulus n=24≥2 and original integer base a=24. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

• G001 — Use integer modulus n=24≥2 and original integer base a=24. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-euler-theorem-number-theory

Step-by-step derivation

source operation=eulerthm-nonunit-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[n, original_base, gcd, theorem_applicable, residue, boundary]

Euler ORIGINAL signed base normalization exact quotient := div(24, 24)

Euler ORIGINAL signed base normalization exact quotient = 1

Euler ORIGINAL signed base normalization floor quotient multiple := mul(1, 24)

Euler ORIGINAL signed base normalization floor quotient multiple = 24

Euler ORIGINAL signed base normalization normalized remainder := sub(24, 24)

Euler ORIGINAL signed base normalization normalized remainder = 0

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=24;modulus=24;exact ratio=1;floor quotient=1

value=q*n+remainder;normalized residue=0 in[0,24);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[24, 24];nonnegative initial pair=[24, 24]

ALL exact Euclidean rows[left,right,q,r]=[[24, 24, 1, 0]];terminal gcd=24

Euler residue representative0:ORIGINAL gcd inputs=[0, 24];nonnegative initial pair=[0, 24]

ALL exact Euclidean rows[left,right,q,r]=[[0, 24, 0, 0]];terminal gcd=24

Euler residue representative1:ORIGINAL gcd inputs=[1, 24];nonnegative initial pair=[1, 24]

ALL exact Euclidean rows[left,right,q,r]=[[1, 24, 0, 1], [24, 1, 24, 0]];terminal gcd=1

Euler residue representative2:ORIGINAL gcd inputs=[2, 24];nonnegative initial pair=[2, 24]

ALL exact Euclidean rows[left,right,q,r]=[[2, 24, 0, 2], [24, 2, 12, 0]];terminal gcd=2

Euler residue representative3:ORIGINAL gcd inputs=[3, 24];nonnegative initial pair=[3, 24]

ALL exact Euclidean rows[left,right,q,r]=[[3, 24, 0, 3], [24, 3, 8, 0]];terminal gcd=3

Euler residue representative4:ORIGINAL gcd inputs=[4, 24];nonnegative initial pair=[4, 24]

ALL exact Euclidean rows[left,right,q,r]=[[4, 24, 0, 4], [24, 4, 6, 0]];terminal gcd=4

Euler residue representative5:ORIGINAL gcd inputs=[5, 24];nonnegative initial pair=[5, 24]

ALL exact Euclidean rows[left,right,q,r]=[[5, 24, 0, 5], [24, 5, 4, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative6:ORIGINAL gcd inputs=[6, 24];nonnegative initial pair=[6, 24]

ALL exact Euclidean rows[left,right,q,r]=[[6, 24, 0, 6], [24, 6, 4, 0]];terminal gcd=6

Euler residue representative7:ORIGINAL gcd inputs=[7, 24];nonnegative initial pair=[7, 24]

ALL exact Euclidean rows[left,right,q,r]=[[7, 24, 0, 7], [24, 7, 3, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative8:ORIGINAL gcd inputs=[8, 24];nonnegative initial pair=[8, 24]

ALL exact Euclidean rows[left,right,q,r]=[[8, 24, 0, 8], [24, 8, 3, 0]];terminal gcd=8

Euler residue representative9:ORIGINAL gcd inputs=[9, 24];nonnegative initial pair=[9, 24]

ALL exact Euclidean rows[left,right,q,r]=[[9, 24, 0, 9], [24, 9, 2, 6], [9, 6, 1, 3], [6, 3, 2, 0]];terminal gcd=3

Euler residue representative10:ORIGINAL gcd inputs=[10, 24];nonnegative initial pair=[10, 24]

ALL exact Euclidean rows[left,right,q,r]=[[10, 24, 0, 10], [24, 10, 2, 4], [10, 4, 2, 2], [4, 2, 2, 0]];terminal gcd=2

Euler residue representative11:ORIGINAL gcd inputs=[11, 24];nonnegative initial pair=[11, 24]

ALL exact Euclidean rows[left,right,q,r]=[[11, 24, 0, 11], [24, 11, 2, 2], [11, 2, 5, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative12:ORIGINAL gcd inputs=[12, 24];nonnegative initial pair=[12, 24]

ALL exact Euclidean rows[left,right,q,r]=[[12, 24, 0, 12], [24, 12, 2, 0]];terminal gcd=12

Euler residue representative13:ORIGINAL gcd inputs=[13, 24];nonnegative initial pair=[13, 24]

ALL exact Euclidean rows[left,right,q,r]=[[13, 24, 0, 13], [24, 13, 1, 11], [13, 11, 1, 2], [11, 2, 5, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative14:ORIGINAL gcd inputs=[14, 24];nonnegative initial pair=[14, 24]

ALL exact Euclidean rows[left,right,q,r]=[[14, 24, 0, 14], [24, 14, 1, 10], [14, 10, 1, 4], [10, 4, 2, 2], [4, 2, 2, 0]];terminal gcd=2

Euler residue representative15:ORIGINAL gcd inputs=[15, 24];nonnegative initial pair=[15, 24]

ALL exact Euclidean rows[left,right,q,r]=[[15, 24, 0, 15], [24, 15, 1, 9], [15, 9, 1, 6], [9, 6, 1, 3], [6, 3, 2, 0]];terminal gcd=3

Euler residue representative16:ORIGINAL gcd inputs=[16, 24];nonnegative initial pair=[16, 24]

ALL exact Euclidean rows[left,right,q,r]=[[16, 24, 0, 16], [24, 16, 1, 8], [16, 8, 2, 0]];terminal gcd=8

Euler residue representative17:ORIGINAL gcd inputs=[17, 24];nonnegative initial pair=[17, 24]

ALL exact Euclidean rows[left,right,q,r]=[[17, 24, 0, 17], [24, 17, 1, 7], [17, 7, 2, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative18:ORIGINAL gcd inputs=[18, 24];nonnegative initial pair=[18, 24]

ALL exact Euclidean rows[left,right,q,r]=[[18, 24, 0, 18], [24, 18, 1, 6], [18, 6, 3, 0]];terminal gcd=6

Euler residue representative19:ORIGINAL gcd inputs=[19, 24];nonnegative initial pair=[19, 24]

ALL exact Euclidean rows[left,right,q,r]=[[19, 24, 0, 19], [24, 19, 1, 5], [19, 5, 3, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative20:ORIGINAL gcd inputs=[20, 24];nonnegative initial pair=[20, 24]

ALL exact Euclidean rows[left,right,q,r]=[[20, 24, 0, 20], [24, 20, 1, 4], [20, 4, 5, 0]];terminal gcd=4

Euler residue representative21:ORIGINAL gcd inputs=[21, 24];nonnegative initial pair=[21, 24]

ALL exact Euclidean rows[left,right,q,r]=[[21, 24, 0, 21], [24, 21, 1, 3], [21, 3, 7, 0]];terminal gcd=3

Euler residue representative22:ORIGINAL gcd inputs=[22, 24];nonnegative initial pair=[22, 24]

ALL exact Euclidean rows[left,right,q,r]=[[22, 24, 0, 22], [24, 22, 1, 2], [22, 2, 11, 0]];terminal gcd=2

Euler residue representative23:ORIGINAL gcd inputs=[23, 24];nonnegative initial pair=[23, 24]

ALL exact Euclidean rows[left,right,q,r]=[[23, 24, 0, 23], [24, 23, 1, 1], [23, 1, 23, 0]];terminal gcd=1

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

Euler complete reduced residue count phi = 8

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

increasing COMPLETE U(n)=[1, 5, 7, 11, 13, 17, 19, 23];phi(n)=8;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

ORIGINAL normalized base=0;COMPLETE totient exponent=8;modulus=24

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

Euler power bit0 base square := mul(0, 0)

Euler power bit0 base square = 0

Euler power bit0 squared-base reduction exact quotient := div(0, 24)

Euler power bit0 squared-base reduction exact quotient = 0

Euler power bit0 squared-base reduction floor quotient multiple := mul(0, 24)

Euler power bit0 squared-base reduction floor quotient multiple = 0

Euler power bit0 squared-base reduction normalized remainder := sub(0, 0)

Euler power bit0 squared-base reduction normalized remainder = 0

Euler power bit0 squared-base reduction:ORIGINAL signed integer=0;modulus=24;exact ratio=0;floor quotient=0

value=q*n+remainder;normalized residue=0 in[0,24);same congruence class,NOT absolute-value replacement

step=0;previous[accumulator,base,exponent]=[1, 0, 8];low bit=0

next[accumulator,base,exponent]=[1, 0, 4];split exponent=2*q+bit preserves original-power invariant

Euler power bit1 base square := mul(0, 0)

Euler power bit1 base square = 0

Euler power bit1 squared-base reduction exact quotient := div(0, 24)

Euler power bit1 squared-base reduction exact quotient = 0

Euler power bit1 squared-base reduction floor quotient multiple := mul(0, 24)

Euler power bit1 squared-base reduction floor quotient multiple = 0

Euler power bit1 squared-base reduction normalized remainder := sub(0, 0)

Euler power bit1 squared-base reduction normalized remainder = 0

Euler power bit1 squared-base reduction:ORIGINAL signed integer=0;modulus=24;exact ratio=0;floor quotient=0

value=q*n+remainder;normalized residue=0 in[0,24);same congruence class,NOT absolute-value replacement

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

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

Euler power bit2 base square := mul(0, 0)

Euler power bit2 base square = 0

Euler power bit2 squared-base reduction exact quotient := div(0, 24)

Euler power bit2 squared-base reduction exact quotient = 0

Euler power bit2 squared-base reduction floor quotient multiple := mul(0, 24)

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

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

Euler power bit2 squared-base reduction normalized remainder = 0

Euler power bit2 squared-base reduction:ORIGINAL signed integer=0;modulus=24;exact ratio=0;floor quotient=0

value=q*n+remainder;normalized residue=0 in[0,24);same congruence class,NOT absolute-value replacement

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

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

Euler power bit3 accumulator product := mul(1, 0)

Euler power bit3 accumulator product = 0

Euler power bit3 accumulator reduction exact quotient := div(0, 24)

Euler power bit3 accumulator reduction exact quotient = 0

Euler power bit3 accumulator reduction floor quotient multiple := mul(0, 24)

Euler power bit3 accumulator reduction floor quotient multiple = 0

Euler power bit3 accumulator reduction normalized remainder := sub(0, 0)

Euler power bit3 accumulator reduction normalized remainder = 0

Euler power bit3 accumulator reduction:ORIGINAL signed integer=0;modulus=24;exact ratio=0;floor quotient=0

value=q*n+remainder;normalized residue=0 in[0,24);same congruence class,NOT absolute-value replacement

Euler power bit3 base square := mul(0, 0)

Euler power bit3 base square = 0

Euler power bit3 squared-base reduction exact quotient := div(0, 24)

Euler power bit3 squared-base reduction exact quotient = 0

Euler power bit3 squared-base reduction floor quotient multiple := mul(0, 24)

Euler power bit3 squared-base reduction floor quotient multiple = 0

Euler power bit3 squared-base reduction normalized remainder := sub(0, 0)

Euler power bit3 squared-base reduction normalized remainder = 0

Euler power bit3 squared-base reduction:ORIGINAL signed integer=0;modulus=24;exact ratio=0;floor quotient=0

value=q*n+remainder;normalized residue=0 in[0,24);same congruence class,NOT absolute-value replacement

step=3;previous[accumulator,base,exponent]=[1, 0, 1];low bit=1

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

remaining exponent0;final normalized accumulator=0

EXACT original normalized base^8 modulo24=0

ACTUAL original n=24,a=24;gcd=24>1;phi(n)=8;computed totient-power residue=0

Euler theorem applicability=No because gcd1 premise fails;retain ACTUAL residue;do NOT cancel a nonunit or infer prime/composite status from this marker

Reverse checks

• S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S060 · premise_and_result_audit — Recompute Euler power bit3 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute Euler power bit3 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Recompute Euler power bit3 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute Euler power bit3 base square with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S055 · premise_and_result_audit — Recompute Euler power bit3 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute Euler power bit3 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Euler power bit3 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute Euler power bit3 accumulator product with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S049 · premise_and_result_audit — Recompute Euler power bit2 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Euler power bit2 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute Euler power bit2 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Euler power bit2 base square with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S043 · premise_and_result_audit — Recompute Euler power bit1 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute Euler power bit1 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute Euler power bit1 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Euler power bit1 base square with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S037 · premise_and_result_audit — Recompute Euler power bit0 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute Euler power bit0 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Euler power bit0 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute Euler power bit0 base square with exact rational arithmetic, not rounded decimals.
• 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 — Recompute Euler complete reduced residue count phi with exact rational arithmetic, not rounded decimals.
• 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

(24, 24, 24, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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.