# Question 1518: Every Fixed Point of the Given Finite Self-Map [Return to catalogue page 8](https://ocme.evemisslab.com/en/questions/page/8/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations/#q-20261013-advance-20261020-r01-discrete-dynamical-system-0068) 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. R0002-01518 · q-20261013-advance-20261020-r01-discrete-dynamical-system-0068 ## Original question and givens Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=13, b=35. 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(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=13, b=35. 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](https://ocme.evemisslab.com/en/mko/mko-discrete-dynamical-system/) ## Step-by-step derivation S001 · impliesBefore X={0 through 9}; T(x)=(13x+(35)) mod 10 After remainder representative always lies in X Conditions: none Fix the full finite state carrier and the same update rule;negative integers also use the standard remainder from0 through n−1. Step identifiers and dependenciesRule: finite_total_map · Uses: G001 S002 · calculationBefore candidate0: a·x := mul(13, 0) After candidate0: a·x = 0 Conditions: none Multiply the original coefficient by the current state;do not clip negative values to0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore candidate0: a·x+b := sum(0, 35) After candidate0: a·x+b = 35 Conditions: none Add the same offset before reducing modulo n. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore reduce 35 modulo 10 After 35=3·10+5 Conditions: none The integer identity together with0≤remainder