Math CentralQuandaries & Queries


Question from Naseer, a student:

I read somewhere that prime number can be a factor of square only if it occurs atleat twice in its square please explian this with examples.

thank you for your time


Very loosely (not a rigorous proof) if a number is a square, say n = ztimesz and a prime p divides it, what do you think about p dividing z?

If p didn't divide z but divides n this would imply 'part' of p divided one of the z's and the 'rest' of p divided the other. However p is pime and can't be broken into parts thus p divides both z's and p2 divides n.

If you're familiar with the canonical representation of an integer n as a product of primes then the truth of your statement is obvious.


(editor's note) This is often called the "Fundamental Theorem of Arithmetic".

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