In what follows all letters represent positive integers Cons

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).

Solution

(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