Track / Stage 5 / 251 of 400 #851 of 2000

Problem 851

AIME late

Number theory Difficulty 5.6 Prove it

Example 2 Let $a=2 t-1$ . (i) If $a \mid 2 n$ , then $a \mid n$ ; (ii) If $2 \mid a b$ , then $2 \mid b$ .

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

Prove that from $a \mid 2 t n$ and $2 t n = a n + n$ , we get $a \mid (2 t n - a n)$ , which means $a \mid n$ . This proves (i). Since $a b = 2 t b - b$ , $b = 2 t b - a b$ , so $2 \mid b$ . This proves (ii).

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