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When the sequence has infinitely many powers of 2

Source: Benelux Mathematical Olympiad 2012

April 23, 2012

floor functionarithmetic sequencenumber theory proposednumber theory

Problem Statement

A sequence

a_1,a_2,\ldots ,a_n,\ldots

of natural numbers is defined by the rule

a_{n+1}=a_n+b_n\ (n=1,2,\ldots)

where

b_n

is the last digit of

a_n

. Prove that such a sequence contains infinitely many powers of

2

if and only if

a_1

is not divisible by

5