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Zero is the last digit of S (Posted on 2018-06-16) Difficulty: 2 of 5
Let S = a^5 + b^5 + c^5 + d^5, and
a,b,c,d are integers fulfilling a+b+c+d=0
Prove that S must be divisible by 10.

No Solution Yet Submitted by Ady TZIDON    
Rating: 5.0000 (1 votes)

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Solution Answer | Comment 1 of 3
For any number x, x^5=x mod 10. We can see this from the 5th powers of 0 to 9.

0^5=0
1^5=1
2^5=32
3^5=243
4^5=1024
5^5=3125
6^5=7776
7^5=16807
8^5=32768
9^5=59049

Then, S=a^5+b^5+c^5+d^5=a+b+c+d mod 10. Therefore, S=0 mod 10, so S is divisible by 10.


  Posted by Math Man on 2018-06-16 19:53:51
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