Is (n^2+3)(n^2+15) divisible by 32 for odd positive integers n?

[sushichan]

Homework Statement

Prove that (n²+3)(n²+15) is divisible by 32 for all odd positive integers n.

Homework Equations

I suppose we are supposed to use mathematical induction since it is in that chapter, but the following questions specifically state that we should use induction but this question doesn't.

The Attempt at a Solution

n=1

(1+3)(1+15)=64=2*32​

n=k

(k²+3)(k²+15)=32A, A∈ℝ​

n=k+1

⇒((k+2)²+3)((k+2)²+15)
= (k²+3)(k²+15) + 8k³+24k²+104k+88
= 32A + 8(k³+3k²+13k+11)​

 

[haruspex]

A∈ℝ

You don't mean that.

n=k+1

Think about that choice again. Note that it says:

all odd positive integers n