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# Help Me Now

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What is the sum of the final three digits of the integer representation of $5^{100}$?

Really need help.

May 9, 2022

#1
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By Euler's THeorem, 5^{100} = 125 mod 1000 so the sum of the three digits is 1 + 2 + 5 = 8.

May 9, 2022
#2
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In fact, it is 625 mod 1000 instead.

MaxWong  May 9, 2022
#3
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5^100 mod 10^10 ==9,306,640,625 - these are the last 10 digits

6  +  2  +  5 ==13

May 9, 2022
#4
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Note that $$\begin{cases}5^{100} \equiv 0 \pmod{125}\\5^{100} = 25^{50} \equiv 1^{50} = 1 \pmod 8\end{cases}$$

Using Chinese Remainder Theorem, the integer x with $$\begin{cases}x \equiv 0 \pmod{125} \\x \equiv 1 \pmod 8\end{cases}$$ is unique mod 1000.

Also, note that $$625 = 5(125) \equiv 0 \pmod{125}$$ and $$625 = 78 \times 8 + 1 \equiv 1 \pmod{8}$$.

That means if $$\begin{cases}x \equiv 0 \pmod{125} \\x \equiv 1 \pmod 8\end{cases}$$, then $$x \equiv 625\pmod{1000}$$.

Then $$5^{100} \equiv 625 \pmod{1000}$$, so the last three digits are 625. The sum of final three digits is 6 + 2 + 5 = 13.

May 9, 2022