Prove or disprove that the fourth power of an odd integer always leaves a remainder of 1 when divided by 16.
We will prove the proposition:
Notice $\displaystyle (2n+1)^4 - 1 = 2n(2n+2)(4n^2+4n + 2) = 8n(n+1)(2n^2+ 2n + 1)$. Then observe that $\displaystyle n(n+1)$ is always even. Thus $\displaystyle 16$ divides $\displaystyle 8n(n+1)$. This means 16 divides $\displaystyle (2n+1)^4 - 1$.