Coprime integers and prime divisors

Forward. If $\gcd(a,b)=1$ and prime $p\mid a,b$, then $p\mid\gcd(a,b)=1$, contradiction.

Converse. If no prime divides both, then any common divisor $d>1$ has a prime divisor $p$ (previous result), which divides both. Contradiction. So $d=1$.

Conclusion. Coprimality is equivalent to disjoint prime divisor sets.

Remark. $\gcd(a,b)=\prod_{p\mid a,b}p^{\min(v_p(a),v_p(b))}$.