Math Help - Powers of Odd Integers

1. Powers of Odd Integers

Prove or disprove that the fourth power of an odd integer always leaves a remainder of 1 when divided by 16.

2. Originally Posted by creativewisdom
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 $(2n+1)^4 - 1 = 2n(2n+2)(4n^2+4n + 2) = 8n(n+1)(2n^2+ 2n + 1)$. Then observe that $n(n+1)$ is always even. Thus $16$ divides $8n(n+1)$. This means 16 divides $(2n+1)^4 - 1$.

3. Originally Posted by Isomorphism
We will prove the proposition:
Notice $(2n+1)^4 - 1 = 2n(2n+2)(4n^2+4n + 2) = 8n(n+1)(2n^2+ 2n + 1)$. Then observe that $n(n+1)$ is always even. Thus $16$ divides $8n(n+1)$. This means 16 divides $(2n+1)^4 - 1$.

Thank you so much for the help. My question, however, is how did you derive this proposition?

4. Originally Posted by creativewisdom
Thank you so much for the help. My question, however, is how did you derive this proposition?
Well... thats what my previous post did...I derived the result. Were you not looking for a proof?