Question

Find all triples of positive integers $a,b,c$ such that $\gcd(a,b,c)=10$ and $\operatorname{lcm}(a,b,c)=100$.

This is a question from An Introduction to Number Theory (Niven et al.), which I was suggested to read by my prof. Any ideas on how to start?

I thought of using $(a,b)\times[a,b]=ab$, but I think that property does not extend to triples, and even for pairs, I wouldn't obtain all solutions.

Answer 1

It's helpful to think of $\gcd$ and $\operatorname{lcm}$ on integers as $\min$ and $\max$, respectively, on the sequence of exponents in their prime factorizations. The given values tell you that among the three numbers the minimum exponents for $2$ and $5$ are $1$ and the maximum exponents are $2$, and the exponents for all other primes are $0$. Now it's straightforward to list all triples of exponent pairs for $2$ and $5$ with those properties.

Answer 2

We must have $a=10x$, $\ b=10y \ $ and $\ c=10z\ $ for some $x,y,z \in \mathbb Z^+$ because the greatest common divisor of $a,b,c$ is $10$.

So, $lcm(10x,10y,10z)=100$. Now you should guess for $x,y$ and $z$.

But make sure that $gcd(x,y,z)=1$.