(i) if the gcd(a,b) = 1, then $\displaystyle p\nmid a \ \ \text{and} \ \ p\nmid b$

$\displaystyle 1|a \ \ \text{and} \ \ 1|b$

$\displaystyle p|(pa) \ \ \text{and} \ \ p|(pb)$

$\displaystyle \Rightarrow p|a \ \ \text{or} \ \ p|p \ \ \text{and} \ \ p|b \ \ \text{or} \ \ p|p$

Since p|p, $\displaystyle p\nmid a \ \ \text{and} \ \ p\nmid b$

(ii) if $\displaystyle p\nmid a \ \ \text{and} \ \ p\nmid b$, then the gcd(a,b) = 1

$\displaystyle p\nmid a, \ \ p\nmid b, \ \ \text{and} \ \ p|p$

$\displaystyle p|(pa) \ \ \text{and} \ \ p|(pb)$

$\displaystyle \Rightarrow 1|a \ \ \text{and} \ \ 1|b$

$\displaystyle \Rightarrow 1=ar+bs \ \ r,s\in\mathbb{Z}$

$\displaystyle \Rightarrow \text{gcd}(a,b)=1$

Correct?