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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