Prove: $$ \text{lcm}(x,y)=\frac{|x\cdot y|}{\text{gcd}(x,y)}$$
I used many ways to do it, all failed. One of them was to represent $|x\cdot y|$ as a sum of primes then $\text{gcd}(x,y)$ as a sum of primes and do the operation but I ended up with a false result since some prime factors were left.
So what is the simplest proof for the gcd to lcm relation?
Let $\gcd(x,y)=d$ and, $x=dx_1$ and $y=dy_1$.
We show that $\text{lcm}(x,y)=dx_1y_1$.
Let $x$ and $y$ divide $m$.
Then $m$ is divisible by $d$.
Write $m=dm_1$.
So, $x_1,y_1|m_1$.
Since $\gcd(x_1,y_1)=1$, there exists integers $a$ and $b$ such that $ax_1+by_1=1$.
This gives $m_1ax_1+m_1ay_1=m_1$.
But $x_1y_1$ divides the LHS and hence the RHS too.
So $x_1y_1$ divides $m_1$.
Thus $dx_1y_1|dm_1$.
Hence $dx_1y_1|m$.
So $m\geq dx_1y_1$.
We have show than any multiple of both $x$ and $y$ is greater than or equal to $dx_1y_1$.
Noting that $x$ and $y$ divide $dx_1y_1$, we conclude that $dx_1y_1$ is the lcm of $x$ and $y$.
From here the rest is trivial.