SOLUTION: What is the remainder when (2^2015+2015^2) when divided by seven

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Question 1072932: What is the remainder when (2^2015+2015^2) when divided by seven
Answer by ikleyn(52817) About Me  (Show Source):
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Let us do it step by step. 

1.  2%5E2015 = %282%5E3%29%5E671%2A2%5E2 = 4%2A8%5E671 = 4%2A%287%2B1%29%5E671.

2.  %287%2B1%29%5E671  gives the remainder 1 when is divided by 7.

3.  Therefore,  4%2A%287%2B1%29%5E671  gives the remainder  4*1 = 4  when is divided by 7.

4.  Thus, the first addend,   2%5E2015,  gives the remainder  4  when is divided by 7.

5.  2015 gives the remainder  2   when is divided by 7.

    So,  2015%5E2  gives the remainder  2*2 = 4   when is divided by 7.

6.  Finally,  the sum  2%5E2015+%2B+2015%5E2  gives the remainder 4 + 4 = 8  when is divided by 7.

    It is the same as to say that the sum  2%5E2015+%2B+2015%5E2  gives the remainder 1  when is divided by 7.

Answer.  The remainder under the question is 1.