GCD $\Rightarrow$ Greatest Common Divisor of $a$ and $b$ : $gcd(a,b)$

If $gcd(a,b) = 1$ then $a$ and $b$ are said to be relatively prime.

If two positive integers $a$ and $b$ have the factorisations : $a=\prod_{p\epsilon \mathbb{P} }^{\infty}{p}^{\alpha_{a} }$ and $b=\prod_{p\epsilon \mathbb{P} }^{\infty}{p}^{\alpha_{b} }$ then $gcd(a,b)=\prod_{p\epsilon \mathbb{P} }^{\infty}{p}^{\alpha_{min(a,b)} }$, where $min(a,b)$ mean the smaller of the two exponents $\alpha_{a}$ and $\alpha_{b}$.

LCM $\Rightarrow$ Least Common Multiple of $a$ and $b$ : $lcm(a,b)$

If two positive integers $a$ and $b$ have the factorisations : $a=\prod_{p\epsilon \mathbb{P} }^{\infty}{p}^{\alpha_{a} }$ and $b=\prod_{p\epsilon \mathbb{P} }^{\infty}{p}^{\alpha_{b} }$ then $lcm(a,b)=\prod_{p\epsilon \mathbb{P} }^{\infty}{p}^{\alpha_{max(a,b)} }$, where $max(a,b)$ mean the bigger of the two exponents $\alpha_{a}$ and $\alpha_{b}$.

$gcd(a,b).lcm(a,b)=a.b$

If two positive integers $a$ and $b$ have the factorisations : $a=\prod_{p\epsilon \mathbb{P} }^{\infty}{p}^{\alpha_{a} }$ and $b=\prod_{p\epsilon \mathbb{P} }^{\infty}{p}^{\alpha_{b} }$ then $a.b=\prod_{p\epsilon \mathbb{P} }^{\infty}{p}^{\alpha_{a}+\alpha_{b} }$.