Prove 5^(2n+1)+11^(2n+1)+17^(2n+1) is divisible by 33.

I wonder if there are other ways that can prove this.

My way of proving this:

Let P(n) be "5^(2n+1)+11^(2n+1)+17^(2n+1) is divisible by 33." for all positive integers n.

When n = 1, 5^3+11^3+17^3 = 33*193

Thus P(1) is true.

Assume P(k) is true for some positive integers k.

i.e. 5^(2k+1)+11^(2k+1)+17^(2k+1) = 33N, where N is an integer.

When n = k+1,

5^[2(k+1)+1] + 11^[2(k+1)+1] + 17^[2(k+1)+1]

=5^(2k+3) + 11^(2k+3) + 17^(2k+3)

=25[5^(2k+1)] + 121[11^(2k+1)] + 289[17^(2k+1)]

=25[5^(2k+1)+11^(2k+1)+17^(2k+1)] + 96[11^(2k+1)] + 264[17^(2k+1)]

=25(33N) + 99[11^(2k+1)] - 3[11^(2k+1)] + 264*17^(2k+1)

=33[25N + 3*11^(2k+1) - 11^(2k) + 8*17^(2k+1)]

Since N and k are integers,

25N + 3*11^(2k+1) - 11^(2k) + 8*17^(2k+1) is an integer.

Therefore, 25N + 3*11^(2k+1) - 11^(2k) + 8*17^(2k+1) is divisible by 33.

Thus P(k+1) is true.

By M.I., P(n) is true for all positive integers n.