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?
 A: 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.
A: Outside of the definition of $gcd(a,b)$ and $lcm(a,b)$, this proof only requires you to know that every integer can be expressed as a unique product of primes. I use a contrapositive proof to prove 
"$\implies$" and a direct proof to prove                     "$\impliedby$"
By contrapositive, assume $gcd(a,b)=d > 1$. Then $a=dr$ and $b=ds$. So $lcm(a,b) \leq drs < (dr)(ds) = ab$
Now assume $gcd(a,b) = 1$. Writing $a$ and $b$ as a product of primes we have:
$a=p_1p_2...p_n$
$b=q_1q_2...q_m$
Where $p$,$q$ are prime and $p_i \neq q_j$ $\forall i,j$ (or else a and b would have a common divisor greater than 1).
Then, any number divisible by a and b must be of the form:
$k(p_1...p_n)(q_1...q_n)$ for some natural number $k$. Then, the smallest such number that is still divisble by a and b is when k=1. So, $lcm(a,b)=ab$.
A: It is known that:
$$gcd(a,b) \cdot lcm(a,b)=a \cdot b $$


*

*IF $gcd(a,b)=1$:
$$ gcd(a,b) \cdot lcm(a,b)=a \cdot b  \Rightarrow 1 \cdot lcm(a,b)=a \cdot b \Rightarrow lcm(a,b)=a \cdot b$$

*IF $lcm(a,b)=a \cdot b$:
$$gcd(a,b) \cdot lcm(a,b)=a \cdot b \Rightarrow gcd(a,b) \cdot a \cdot b=a \cdot b \Rightarrow gcd(a,b)=1 $$

A: If the highest power of prime $p,$ in $a,b$ be $A\ge0,B\ge0$
gcd$\displaystyle(a,b)=\prod p^{\text{min}(A,B)}$; lcm $\displaystyle(a,b)=\prod p^{\text{max}(A,B)}$
$\displaystyle ab=\prod p^{A+B}$
We need $A+B=\text{max}(A,B)$ for each prime $p$ that divides $ab$
But we know $A+B=\text{max}(A,B)+\text{min}(A,B)$
$\displaystyle\implies \text{min}(A,B)=0\implies$ at least one of $A,B$ must be $0$
$\displaystyle\implies$gcd$(a,b)=1$
A: In this answer, it is shown that
$$
\gcd(a,b)\,\mathrm{lcm}(a,b)=ab
$$
From here it is pretty simple to show that $\mathrm{lcm}(a,b)=ab$ if and only if $\gcd(a,b)=1$.
