Competition problem — Number theory

Library / /74 of 377

Number theory Difficulty 4.7 AIME Prove it United States

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

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: $50$

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

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