The question states prove,(adsbygoogle = window.adsbygoogle || []).push({});

If p is prime and p | a^n then p^n | a^n

I am pretty sure I have i just may need someone to help clean it up.

There are two relevant theorems i have for this.

the first says p is prime if and if p has the property that if p | ab then p | a or p | b

the second one is that if p is prime and p | a1a2a3.....an, then p must divide one of the a_i.

so for the proof i am assuming p | a^n which i can rewrite as

p | a*a*a.......an-1*an. so this is saying p(q) = a*a*a....an-1*an for some integer q.

now if I look at p^n | a^n thats the same as

p*p*p.....pn-1*pn | a*a*a.....an-1*an

well p(p*p*p....pn-1*p) | a*a*a....an-1*an

is that the way to go?

Or maybe before when i had that p(q) = a*a*a....an-1*an for some integer q.

just set q = p^n-1 so that p(q) = p^n.

I feel like the later way should do it.

Is this right?

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# Homework Help: Abstract algebra proof involving prime numbers

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