12 Days of Christmas GMAT Competition - Day 5: x and y are positive in

Official Solution:


\(x\) and \(y\) are positive integers and the sum of the digits of \(x*y\) is 1. If \(x\) and \(y\) are not multiples of 10, which of the following cannot be the remainder when \(|x - y|\) is divided by 10?

I. 2

II. 4

III. 5


A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II and III


The sum of the digits of \(x*y\) is 1 means that \(x*y\) can be 10, 100, 1,000, 10,000, ... All these numbers have only 2 and 5 as primes, so those will also be the only primes of \(x\) and \(y\). But since we know that \(x\) and \(y\) are not multiples of 10, then \(x\) and \(y\) cannot have both 2 and 5 as their primes, meaning that \(x\) must have only 2's and \(y\) must have equal number of 5's, or vise-versa. So, \(x=2^n\) and \(y=5^n\), or vise-versa (\(n\) is a positive integer).

The question asks to find which of the options cannot be the remainder when \(|x - y|=5^n-2^n\) is divided by 10. When a number is divided by 10, the remainder is always the units digit of that number, so we need to find which numbers could be the units digit of \(5^n-2^n\).

The units digit of \(5^n\) is always 5.

The units digit of \(2^n\) can be 2, 4, 8, or 6.

Thus, the units digit of \(5^n-2^n\) can be \(...5-...2=...3\), \(...5-...4=...1\), \(...5-...8=...7\), or \(...5-...6=...9\).

So, the remainder when \(|x - y|\) is divided by 10 can only be 3, 1, 7, or, 9.


Answer: E