+0  
 
0
120
4
avatar

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

Really need help.

 May 9, 2022
 #1
avatar
0

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
avatar+9461 
0

In fact, it is 625 mod 1000 instead.

MaxWong  May 9, 2022
 #3
avatar
+1

5^100 mod 10^10 ==9,306,640,625 - these are the last 10 digits

 

6  +  2  +  5 ==13

 May 9, 2022
 #4
avatar+9461 
0

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

18 Online Users

avatar
avatar