$$\operatorname{lcm}(a, b, c) = 100$$
What are all the triads $a, b, c $ that happen?
How many answers are there?
We've the canonical form $100=2^2 \cdot 5^2$ therefore $\tau(100)=9$ Assuming $a,b,c $ to be distinct integers there are $\binom{18}{3}$ tuples $(a,b,c)$ satisfying the desired conditions Sidenote : OP must specify the nature of $a,b,c$, i.e. whether $a,b,c \in \mathbb{Z}$or $\mathbb{Z}^{+}$ and whether they're all distinct or not ??? or I think the answer is $\binom{3}{1}^2 \cdot 9^2 $.
Who knows if it is right or not please help prove the solution ?
The number of ordered triples of nonnegative integers less than or equal to $2$ with at least one coordinate equal to $2$ (and hence having a maximum of $2$) is $3^3-2^3=27-8=19$ (all ordered triples, minus those with all coordinates less than $2$).
Hence, since LCMs are all about taking the maximum of the exponents of each prime factor (and likewise, the minimum for the GCD), the answer is $19^2=361$, assuming that only solutions with $a, b,$ and $c$ all positive are allowed. If one allows at least one of $a, b,$ and $c$ to be negative, then one must multiply the answer by $8$ to get $361 \cdot 8=2888$.