Competition problem — Number theory

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Number theory Difficulty 5.7 AIME, harder Prove it

1. Prove that when the unit digit of any integer $a$ is divisible by 2, then this integer is a multiple of 2.

Solution

1. Proof: Any integer $a$ can be written as
$a=10 n+b$

where $n$ is an integer and $0 \leqslant b<10$ . Since $2 \mid 10$ , (1) and the assumption $2 \mid b$ , it follows that $2 \mid a$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.