Show that if $d_1e_1=d_2e_2,\ and\ \gcd(e_1,e_2)=1\Rightarrow\ lcm(d_1,d_2)=d_1e_1=d_2e_2$

Question

Show that for positive integers if $d_1e_1=d_2e_2\ and \gcd(e_1,e_2)=1\Rightarrow\ lcm(d_1,d_2)=d_1e_1=d_2e_2$
We know that: $$lcm(d_1,d_2)gcd(d_1,d_2)=d_1d_2$$ but this doesn't help much!
What's the trick?

Answer 1

Let $k=\gcd(d_1,d_2)$. Then $d_1=kd_1'$ and $d_2=kd_2'$ where $d_1'$ and $d_2'$ are relatively prime. Thus from the relationship you stated we have $$\text{lcm}(d_1,d_2)k=k^2d_1'd_2',$$ and therefore $$\text{lcm}(d_1,d_2)=kd_1'd_2'.$$ It remains to show that $kd_1'd_2'=d_1e_1=kd_1'e_1$, or equivalently $d_2'=e_1$.

We were told that $d_1e_1=d_2e_2$, or equivalently that $d_1'e_1=d_2'e_2$. Since $e_1$ and $e_2$ are relatively prime, we have $e_1\mid d_2'$. And because $d_1'$ and $d_2'$ are relatively prime, we have $d_1'\mid e_2$.

But because $d_1'e_1=d_2'e_2$, it follows that $e_1=d_2'$ and $d_1'=e_2$.