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This is taken from an answer book that I have. I don't understand the bolded step. Can someone explain it to me?

Suppose x = p/q where p and q are natural numbers with no common factor. Then:

p^{n}/q^{n}+ a_{n-1}p^{n-1}/q^{n-1}+ ... + a_{o}= 0

and multiplying both sides by q^{n}gives

p^{n}+ a_{n-1}p^{n-1}q + ... + a_{o}q^{n}= 0

Now if q ≠ ±1 then q has some prime number as a factor. This prime number divides every term of the second equation other than p^{n},so it must divide pTherefore it divides p, a contradiction. So q = ±1, which means that x is an integer.^{n}also.

Once again, it's the bolded step that I don't understand. Why must it divide p^{n}?

Thanks in advance.

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# Homework Help: Prove that x is irrational unless it is an integer.

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