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Zero is the last digit of S (Posted on 2018-06-16)

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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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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