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

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