Show that lcm$(a,b)= ab$ if and only if gcd$(a,b)=1$ Not sure how to begin. If gcd$(a,b)=1$ what can I deduce from that?

Hint $ $ note that $n\mapsto ab/n\,$ bijects the common divisors of $\,a,b\,$ with the common multiples $\le ab.$ Being order-reversing it maps the greatest common divisor to the least common multiple, i.e. $\,\gcd(a,n)\mapsto ab/\gcd(a,b) = {\rm lcm}(a,b).\,$ Thus $\,\gcd(a,b)=1\iff {\rm lcm}(a,b) = ab$.
Remark $\ $ For more on the (involution) duality between gcd and lcm see here and here.
