Let n be a positive odd integer. Prove that n4 ≡ 1 (mod 16). You may use any of the mod rules provided in class (In addition, if you end up getting an expression of the form a (3a+1) somewhere in your work where a is an integer, separately end up proving that this expression is always even and use that result to complete your proof).

If is odd, then we can write for some .

Expanding gives

This reduces to

Notice that is 8 times the product of two consecutive integers, one even and the other odd. This product is even and thus divisible by 2, so that where , and the equivalence further reduces to

as required.

fist on the left goes with the third on the right. second on the left goes with the fourth on the right. third on the left goes with the second on the right. the fourth on the left goes with first on the right

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