- #1
Math100
- 768
- 219
- 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 ##.
Last edited by a moderator: