Thread: Proof by induction stuck on a sum help needed

Code:

Given that m is an odd positive integer, prove that $\displaystyle (m^2+3)(m^2+15)$ is divisible by 32 for all such values of m.

I made it divisible by 8:
$\displaystyle ((k+2)^2+3)((k+2)^2+15)-((k^2+3)(k^2+15))
=8 (k^3+3k^2+13k+11)$
But I could not make it divisible by 32. A little help please, I feel I was very close to the solution.
My method is right, right?

Hello,

Originally Posted by ssadi

Code:

Given that m is an odd positive integer, prove that $\displaystyle (m^2+3)(m^2+15)$ is divisible by 32 for all such values of m.

I made it divisible by 8:
$\displaystyle ((k+2)^2+3)((k+2)^2+15)-((k^2+3)(k^2+15))
=8 (k^3+3k^2+13k+11)$
But I could not make it divisible by 32. A little help please, I feel I was very close to the solution.
My method is right, right?

You can use the fact that k is odd, that is to say $\displaystyle k=2n+1$

It'll give you what you want, but the method you use looks very long to me...
Dunno if there's another way (thinking on it)