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Problem 879

AMC 12 late, AIME early

Number theory Difficulty 4.7 Prove it Harvard-MIT Mathematics Tournament · United States

Find the smallest positive integer $n$ such that $107 n$ has the same last two digits as $n$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Solution:
The two numbers have the same last two digits if and only if $100$ divides their difference $106 n$ , which happens if and only if $50$ divides $n$ .

Answer: $50$

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.