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

AI Thread Summary
The discussion revolves around proving that the expression (n^2+3)(n^2+15) is divisible by 32 for all odd positive integers n. Participants suggest using mathematical induction, although the original problem does not explicitly require it. The initial case for n=1 shows that the expression equals 64, which is divisible by 32. The induction step involves evaluating the expression for n=k and n=k+1, leading to a formula that includes a term divisible by 32. The conversation emphasizes the importance of correctly applying the induction hypothesis to ensure the proof holds for all odd positive integers.
sushichan
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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)​
 
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sushichan said:
A∈ℝ
You don't mean that.
sushichan said:
n=k+1
Think about that choice again. Note that it says:
sushichan said:
all odd positive integers n
 

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