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Getting the right count (Posted on 2015-07-14)

Find the total number of positive integers n not over 2015 such that n^4+5n^2+9 is divisible by 5.

Source: SMO, slightly modified

No Solution Yet Submitted by Ady TZIDON

Rating: 3.0000 (1 votes)

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Analytical Solution

Comment 1 of 1

Well, we can ignore the 5n^2 term, which is always divisible by 5.

All we need is n^4 = 1 mod 5.

1^4 = 1 mod 5

2^4 = 16 = 1 mod 5

3^4 = 81 = 1 mod 5

4^4 = (2^4)^2 = 1 mod 5

5^4 = 0 mod 5

so this works for 4/5 of all integers.

Answer = 2015 * 4/5 = 1612

Posted by Steve Herman on 2015-07-14 10:05:06

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