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| Math Central | Quandaries & Queries |
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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 |
Hello,
Very loosely (not a rigorous proof) if a number is a square, say n = z
z 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†.
Penny
†(editor's note) This is often called the "Fundamental Theorem of Arithmetic".
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