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 -

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2021 JMPSC Sprint Problems/Problem 15

Contents

Problem

Solution

Solution 2

Solution 3

See also