proof if gcd(a,b)=1 and a|c b|c then ab|c

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That follows pretty quickly from the definition of "greatest common dominator", doesn't it? If gcd(a, b)= 1, then a and b have no common factors. You might try, for one thing, consider the prime factorization of c.

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Originally Posted by dave52

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little rigorous proof:

if and there exists some integers and such that
now as , we can write: (after multiplying both sides by )
after substituting the right expressions for we get , which in other words mean