MathDB
computer divides with 18

Source: Argentina 2000 OMA L3 p4

May 12, 2024
number theory

Problem Statement

A computer program generates a sequence of numbers with the following rule: the first number is written by Camilo; thereafter, the program calculates the integer division of the last number generated by 1818; thus obtains a quotient and a remainder. The sum of that quotient plus that remainder is the next number generated. For example, if Camilo's number is 5291,5291, the computer makes 5291=293×18+175291 = 293 \times 18 + 17, and generates 310=293+17310 = 293 + 17. The next number generated will be 2121, since 310=17×18+4310 = 17 \times 18 + 4 and 17+4=2117 + 4= 21; etc Whatever Camilo's initial number is, from some point on, the computer always generates the same number. Determine what is that number that will be repeated indefinitely, if Camilo's initial number is equal to 2110.2^{110}.