Prove these are relatively prime. (involving Greatest Common Divisor) 
I know that we have to use prime factorization.
$\gcd(\alpha,\beta)$ => using minimum
$\text{lcm}(\alpha,\beta)$ => using maximum
$$\gcd(\alpha,\beta) \times \text{lcm}(\alpha,\beta) = a * b$$
How do I continue or any tips?
Thank You.

One can give a simple proof using Bezout's Theorem. Let $d=\gcd(a,b)$. Then there exist integers $x$ and $y$ such that $ax+by=d$. Thus $da'x+db'y=d$, and therefore $a'x+b'y=1$. Relative primality follows.
Another way: (better) Let $d=\gcd(a,b)$. Then $a=a'd$ and $b=b'd$. If $k\gt 1$ divides both $a'$ and $b'$, then $kd$ divides both $a$ and $b$, contradicting the fact that $d$ is the greatest common divisor of $a$ and $b$.
