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 $100...00 - 25 = 99...99 - 24 = 999...\boxed{75}$.

~Mathdreams

Solution 3

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

- kante314 -

See also

  1. Other 2021 JMPSC Sprint Problems
  2. 2021 JMPSC Sprint Answer Key
  3. All JMPSC Problems and Solutions

The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition. JMPSC.png

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