Question 1514: Every Fixed Point of the Given Finite Self-Map

Original question and givens

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=9, b=27. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

• G001 — Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=9, b=27. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

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-discrete-dynamical-system

Step-by-step derivation

X={0 through 5}; T(x)=(9x+(27)) mod 6

remainder representative always lies in X

candidate0: a·x := mul(9, 0)

candidate0: a·x = 0

candidate0: a·x+b := sum(0, 27)

candidate0: a·x+b = 27

reduce 27 modulo 6

27=4·6+3

T(0)=3

fixed-point condition T(x)=x: No

candidate1: a·x := mul(9, 1)

candidate1: a·x = 9

candidate1: a·x+b := sum(9, 27)

candidate1: a·x+b = 36

reduce 36 modulo 6

36=6·6+0

T(1)=0

fixed-point condition T(x)=x: No

candidate2: a·x := mul(9, 2)

candidate2: a·x = 18

candidate2: a·x+b := sum(18, 27)

candidate2: a·x+b = 45

reduce 45 modulo 6

45=7·6+3

T(2)=3

fixed-point condition T(x)=x: No

candidate3: a·x := mul(9, 3)

candidate3: a·x = 27

candidate3: a·x+b := sum(27, 27)

candidate3: a·x+b = 54

reduce 54 modulo 6

54=9·6+0

T(3)=0

fixed-point condition T(x)=x: No

candidate4: a·x := mul(9, 4)

candidate4: a·x = 36

candidate4: a·x+b := sum(36, 27)

candidate4: a·x+b = 63

reduce 63 modulo 6

63=10·6+3

T(4)=3

fixed-point condition T(x)=x: No

candidate5: a·x := mul(9, 5)

candidate5: a·x = 45

candidate5: a·x+b := sum(45, 27)

candidate5: a·x+b = 72

reduce 72 modulo 6

72=12·6+0

T(5)=0

fixed-point condition T(x)=x: No

all 6 carrier elements audited

complete fixed set=[]

Reverse checks

• 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 — Recompute candidate5: a·x+b with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute candidate5: a·x with exact rational arithmetic, not rounded decimals.
• 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 — Recompute candidate4: a·x+b with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute candidate4: a·x with exact rational arithmetic, not rounded decimals.
• 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 — Recompute candidate3: a·x+b with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute candidate3: a·x with exact rational arithmetic, not rounded decimals.
• 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 — Recompute candidate2: a·x+b with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute candidate2: a·x with exact rational arithmetic, not rounded decimals.
• 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 — Recompute candidate1: a·x+b with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute candidate1: a·x with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Recompute candidate0: a·x+b with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute candidate0: a·x 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

(6, 9, 27, [])

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.