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- Homework Statement
- Given that ##p## is a prime and ##p\mid a^n ##, prove that ## p^n \mid a^n ##.

- Relevant Equations
- None.

Proof:

Suppose that p is a prime and ##p \mid a^n ##.

Note that a prime number is a number that has only two factors,

1 and the number itself.

Then we have (p*1)##\mid##a*## a^{(n-1)} ##.

Thus p##\mid##a, which implies that pk=a for some k##\in\mathbb{Z}##.

Now we have ## a^n ##=## (pk)^n ##

=## p^n k^n ##.

This means ##p^n \mid a^n ##.

Therefore, given that p is a prime and ##p \mid a^n ##,

we have proven that ##p^n \mid a^n ##.

Suppose that p is a prime and ##p \mid a^n ##.

Note that a prime number is a number that has only two factors,

1 and the number itself.

Then we have (p*1)##\mid##a*## a^{(n-1)} ##.

Thus p##\mid##a, which implies that pk=a for some k##\in\mathbb{Z}##.

Now we have ## a^n ##=## (pk)^n ##

=## p^n k^n ##.

This means ##p^n \mid a^n ##.

Therefore, given that p is a prime and ##p \mid a^n ##,

we have proven that ##p^n \mid a^n ##.

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