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## Comment on

12 is a divisor of n## what about 108?

## 108 = (2)(2)(3)(3)(3) = (2^2)

108 = (2)(2)(3)(3)(3) = (2^2)(3^3)

If n = (g^h)(h^g), there's no way to assign values to g and h to get (2^2)(3^3).

IF the question were worded so that n = (g^g)(h^h), then 108 would work if we let g = 2 and h = 3. However, the question tells us that n = (g^h)(h^g), in which case there's no way to assign values to g and h to get (2^2)(3^3).

## Brilliant Explanation. Thanks

## What if you chose 720 for n?

## n cannot equal 720.

n cannot equal 720.

To understand why, let's first examine the prime factorization of 720

n = 720

= (2)(2)(2)(2)(3)(3)(5)

= (2^4)(3^2)(5^1)

We're told that n = (g^h)(h^g), where g and h are PRIME

There's no way to assign values to g and h to get (g^h)(h^g)

= (2^4)(3^2)(5^1).

As such, n cannot equal 720

In fact, n cannot equal any number other than 72

Does that help?

Cheers,

Brent

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