# Difference between revisions of "2021 JMPSC Sprint Problems/Problem 15"

## Problem

Find the last two digits of $10^{10}-5^{10}.$

## Solution

Note that $10^{10}\equiv0\pmod{100}$ and $5^{10}\equiv25\pmod{100}$.

$0-25=-25$. $-25\equiv\boxed{75}\pmod{100}$

## Solution 2

By multiplying out several powers of $5$, we can observe that the last $2$ digits are always $25$ (with the exception of $5^n$ where $n \le 1$). Also, $10^10$ ends with several zeros, so the answer is $100...00 - 25 = 99...99 - 24 = 999...75$.

~Mathdreams

## Solution 3

$$100^{10} \equiv 0 \mod 100$$$$5^{10} \equiv 25 \mod 100$$Therefore, the answer is $75$

- kante314 -