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What is the remainder of the following congruence:
[1^1000 + 2^1000 + 3^1000 +..................+100^1000] mod 1000 =? Any help would be appreciated. Thank you.

Guest Nov 27, 2018
 #1
avatar+701 
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This is a really tedious number theory problem. I would probably try to find a pattern in the mod 1000 of the powers of 2, 3, 4, ect. Remember that any multiple of 10 to the power of 1000 (\(10^{1000}\), \(20^{1000}\), ect.) mod 1000 is 0, which means it can be divided by 1000. Other than that, I think there is just pure computations for this problem.

PartialMathematician  Nov 28, 2018
 #2
avatar+701 
+4

I'll probably feed the data into a computer program to determine the answer.

- PM

PartialMathematician  Nov 28, 2018
 #3
avatar+3197 
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It's a bit more clever than that.

 

If you look at k^1000 % 1000 you'll see it's a periodic sequence with period 10

 

I'm no number theory guru but I think the secret lies in the common factors of k and 1000.

 

I'll post if I figure it out.

Rom  Nov 28, 2018
 #4
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I don't know if there is a shortcut to it, but computers can easily handle it. It is 2001-digit long. 
It begins with......10000431729.......and ends in....  5137781330 mod 1000 =330 

Guest Nov 28, 2018

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