1) By definition, means there is a positive integer n such that . You can prove by contradiction that n is the square of an integer, so that and and since z and d are positive and you can choose m positive, z=md, which is the definition of .
2) If you extend the previous proof by a minor step, you can get gcd(x,y)=gcd(x,y,z) (Prove that a divisor of one is a divisor of the other, in both directions). And a similar argument would apply for gcd(x,z) and gcd(y,z).
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