MathDB

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

18

; 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,

the computer makes

5291 = 293 \times 18 + 17

, and generates

310 = 293 + 17

. The next number generated will be

21

, since

310 = 17 \times 18 + 4

and

17 + 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

2^{110}.

5

Problems(1)