The question is this:
Find all triplets of positive integers $a,b,c$ satisfying $(a,b,c) = 10$ and $\left[a,b,c\right] = 100$ simultaneously. Here, $(x,y)$ is the greatest common divisor of $x$ and $y$ and $[x,y]$ is the least common multiple of $x$ and $y$.
It would be very easy if the question was in $2$ variables, because there is a relation between product of the numbers, the GCD and the LCM, but I am not aware of any relation between them when $3$ variables are involved.
For attempting purpose, one can set $a = 10p$, $b = 10q$ and $c = 10r$, where $(p,q,r) = 1$.
Then $[p,q,r] = 10$. Now I don't have an idea to proceed from here.
I am pretty much a beginner to elementary number theory, so might be I have missed something obvious.
Thanks for solutions!