# 1971 Canadian MO Problems/Problem 7

## Problem

Let be a five digit number (whose first digit is non-zero) and let be the four digit number formed from n by removing its middle digit. Determine all such that is an integer.

## Solution

Let and , where , , , , and are base-10 digits and . If is an integer, then , or

This implies that

Clearly we have that , as is positive. therefore must be equal to 9, and

This simplifies to . The only way that this could happen is that . Then . Therefore the only values of such that is an integer are multiples of 1000.