Question 20081: Multiplication by a Unit Permutes the Complete Reduced Residues

Original question and givens

Use integer modulus n=10≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

• G001 — Use integer modulus n=10≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (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

Step-by-step derivation

source operation=eulerthm-unit-permutation;original snapshot and mathematics retained

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

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

Euler ORIGINAL signed base normalization exact quotient = 7/10

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

Euler ORIGINAL signed base normalization floor quotient multiple = 0

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

Euler ORIGINAL signed base normalization normalized remainder = 7

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

value=q*n+remainder;normalized residue=7 in[0,10);same congruence class,NOT absolute-value replacement

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

ALL exact Euclidean rows[left,right,q,r]=[[7, 10, 0, 7], [10, 7, 1, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal gcd=1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

ALL exact Euclidean rows[left,right,q,r]=[[7, 10, 0, 7], [10, 7, 1, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal gcd=1

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

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

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

ALL exact Euclidean rows[left,right,q,r]=[[9, 10, 0, 9], [10, 9, 1, 1], [9, 1, 9, 0]];terminal gcd=1

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

Euler complete reduced residue count phi = 4

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

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

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

Euler ACTUAL base-unit inverse product = 21

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

Euler ACTUAL inverse check exact quotient = 21/10

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

Euler ACTUAL inverse check floor quotient multiple = 20

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

Euler ACTUAL inverse check normalized remainder = 1

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

value=q*n+remainder;normalized residue=1 in[0,10);same congruence class,NOT absolute-value replacement

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

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)

Euler ORIGINAL a times unit representative1 := mul(7, 1)

Euler ORIGINAL a times unit representative1 = 7

Euler unit representative1 raw image exact quotient := div(7, 10)

Euler unit representative1 raw image exact quotient = 7/10

Euler unit representative1 raw image floor quotient multiple := mul(0, 10)

Euler unit representative1 raw image floor quotient multiple = 0

Euler unit representative1 raw image normalized remainder := sub(7, 0)

Euler unit representative1 raw image normalized remainder = 7

Euler unit representative1 raw image:ORIGINAL signed integer=7;modulus=10;exact ratio=7/10;floor quotient=0

value=q*n+remainder;normalized residue=7 in[0,10);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative3 := mul(7, 3)

Euler ORIGINAL a times unit representative3 = 21

Euler unit representative3 raw image exact quotient := div(21, 10)

Euler unit representative3 raw image exact quotient = 21/10

Euler unit representative3 raw image floor quotient multiple := mul(2, 10)

Euler unit representative3 raw image floor quotient multiple = 20

Euler unit representative3 raw image normalized remainder := sub(21, 20)

Euler unit representative3 raw image normalized remainder = 1

Euler unit representative3 raw image:ORIGINAL signed integer=21;modulus=10;exact ratio=21/10;floor quotient=2

value=q*n+remainder;normalized residue=1 in[0,10);same congruence class,NOT absolute-value replacement

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

Euler ORIGINAL a times unit representative7 = 49

Euler unit representative7 raw image exact quotient := div(49, 10)

Euler unit representative7 raw image exact quotient = 49/10

Euler unit representative7 raw image floor quotient multiple := mul(4, 10)

Euler unit representative7 raw image floor quotient multiple = 40

Euler unit representative7 raw image normalized remainder := sub(49, 40)

Euler unit representative7 raw image normalized remainder = 9

Euler unit representative7 raw image:ORIGINAL signed integer=49;modulus=10;exact ratio=49/10;floor quotient=4

value=q*n+remainder;normalized residue=9 in[0,10);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative9 := mul(7, 9)

Euler ORIGINAL a times unit representative9 = 63

Euler unit representative9 raw image exact quotient := div(63, 10)

Euler unit representative9 raw image exact quotient = 63/10

Euler unit representative9 raw image floor quotient multiple := mul(6, 10)

Euler unit representative9 raw image floor quotient multiple = 60

Euler unit representative9 raw image normalized remainder := sub(63, 60)

Euler unit representative9 raw image normalized remainder = 3

Euler unit representative9 raw image:ORIGINAL signed integer=63;modulus=10;exact ratio=63/10;floor quotient=6

value=q*n+remainder;normalized residue=3 in[0,10);same congruence class,NOT absolute-value replacement

COMPLETE original increasing U(n)=[1, 3, 7, 9];actual images in ORIGINAL unit order=[7, 1, 9, 3]

SOURCE output sorts the mapped list numerically=[1, 3, 7, 9];full-list equality=Yes;not a claim each point is fixed

Reverse checks

• 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 unit representative9 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute Euler unit representative9 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute Euler unit representative9 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative9 with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S038 · premise_and_result_audit — Recompute Euler unit representative7 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute Euler unit representative7 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute Euler unit representative7 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative7 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S033 · premise_and_result_audit — Recompute Euler unit representative3 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Euler unit representative3 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Euler unit representative3 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative3 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S028 · premise_and_result_audit — Recompute Euler unit representative1 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Euler unit representative1 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Euler unit representative1 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative1 with exact rational arithmetic, not rounded decimals.
• 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 — Recompute Euler ACTUAL inverse check normalized remainder with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Euler ACTUAL inverse check floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Euler ACTUAL inverse check exact quotient with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Euler ACTUAL base-unit inverse product with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Recompute Euler complete reduced residue count phi with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute Euler ORIGINAL signed base normalization normalized remainder with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Euler ORIGINAL signed base normalization floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Euler ORIGINAL signed base normalization exact quotient with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([1, 3, 7, 9], [1, 3, 7, 9], 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.