In what follows, all letters represent positive integers. Construct proofs which

Question

In what follows, all letters represent positive integers. Construct proofs which use the Fundamental Theorem of Arithmetic. Keep in mind that a consequence of the Fundamental Theorem is that if a = Pi pi^ei is the prime factorization of a, and b = Pi pi^f_i is the prime factorization of b, then gcd (a, b) = Pi pi^min{e_i, f_i}. Prove that if k is not a square then squareroot k is irrational. Consider k = m^2/n^2 for some m, n and obtain a contradiction. Prove that If ab is a square and gcd(a, b) = 1 then a and b are squares. Prove that if m/n is a fraction in lowest terms and m/n = r/s there exists an integer k such, that r = km and s = kn. Let a, b, n be positive integers. Prove that gcd(a, b) = 1 if and only if gcd (a^n, b^n) = 1. Prove that if a^3|b^2 then a |b. Prove that if c|ab then c|gcd(a, c) middot gcd(b, c).

Explanation / Answer

(4)

We know that gcd(a,bc)=1 if and only if gcd(a,b)=1 and gcd(a,c)=1

At b=c, we get:

gcd(a,b) = 1 if and only if gcd(a,b2) = 1 (Statement 1)

Similarly, gcd(ad,b2) = 1 if and only if gcd(a,b2) = 1 and gcd(d,b2) = 1

At a=d, we get:

gcd(a,b2) = 1 if and only if gcd(a2,b2) = 1 (Statement 2)

On combining statements 1 and 2, we get:

gcd(a,b) = 1 if and only if gcd(a2,b2) = 1

Similarly,

gcd(a,b) = 1 if and only if gcd(an,bn) = 1

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