Two $\displaystyle \mathbb{Z}^+$ are relatively prime iff. their LCM=ab.

$\displaystyle LCM(a,b)=\frac{ab}{GCD(a,b)}$

1. Assume a and b are relatively prime.

$\displaystyle GCD(a,b)=1$

$\displaystyle LCM(a,b)=\frac{ab}{1}=ab$

2. Assume $\displaystyle LCM=ab$.

$\displaystyle LCM(a,b)=\frac{ab}{GCD(a,b)}\rightarrow GCD(a,b)=\frac{ab}{LCM(a,b)} \ \mbox{but since the LCM=ab} \ GCD(a,b)=1$