Given that $$a\cdot b=gcd(a,b)\cdot lcm(a,b)$$
How can we find all the integer solutions $(a,b)$ if $gcd(a,b)=6$ and $lcm(a,b)=540$?
The first thing I did was factorizing using the fundamental theorem of arithmetic.
$$a\cdot b=gcd(a,b)\cdot lcm(a,b)=6\cdot540=2^3\cdot3^4\cdot5$$
I also know that $$gcd(a,b)=p_1^{min(a_1,b_1)}\cdot p_2^{min(a_2,b_2)}\dots p_n^{min(a_n,b_n)}$$ $$lcm(a,b)=p_1^{max(a_1,b_1)}\cdot p_2^{max(a_2,b_2)}\dots p_n^{max(a_n,b_n)}$$ where $a_1\dots a_n$ and $b_1\dots b_n$ are the exponents in the factorization of $a$ and $b$ respectively.
From there I am not sure where to go to obtain all the possible integers $a$ and $b$.
There is already a similar question on Mathematics SE, however the accepted solution is simply a hint towards the correct procedure to do it, while this question already acknowledged that hint.
