Track / Stage 5 / 63 of 400 #663 of 2000

Problem 663

AIME late

Number theory Difficulty 5.2 Find the answer

[ [decimal number system ] $[$ equations in integers $]$

The number $x$ is such that $x^{2}$ ends in 001 (in the decimal number system).

Find the last three digits of the number $x$ (list all possible options).

Your answer

Official solution

$(x-1)(x+1)=x^{2}-1$ is divisible by $1000=2^{3} \cdot 5^{3}$ . Therefore, both numbers $x-1$ and $x+1$ are even (then one of them is divisible by 4, and their product is divisible by 8), and one of them is divisible by 125, and thus by 250.

## Answer

$001,249,251,499,501,749,751,999$ .

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