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jcoughlin

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## Homework Statement

Claim: If n is a positive integer, the prime factorization of 2

^{2n}* 3

^{n}- 1 includes 11 as one of the prime factors.

## Homework Equations

Factor Theorem: a polynomial f(x) has a factor (x-k) iff f(k)=0.

## The Attempt at a Solution

First, we show that (x-1) is a factor of (x

^{n}-1). Let f(x)=x

^{n}-1, and k=1; then f(k)=0, and thus by the factor theorem (x-1) is a factor of (x

^{n}-1).

Next, consider 2

^{2n}* 3

^{n}- 1 rewritten as 12

^{n}-1. As previously demonstrated, x-1 is a factor of x

^{n}-1. Letting x=12, we see that (12-1)=11 is a factor of 12

^{n}-1 for n>0.

Is this sufficient? Or do I need to go further than proving 11|(12

^{n}-1) to show that the prime factorization of 2

^{2n}* 3

^{n}- 1 includes 11 as one of the prime factors?

Thanks,

James

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