GCD and LCM of three numbers Given two positive integers G and L, could you tell me how many solutions of (x, y, z) there are, satisfying that gcd(x, y, z) = G and lcm(x, y, z) = L? gcd(x, y, z) means the greatest common divisor of x, y and z, while lcm(x, y, z) means the least common multiple of x, y and z. Also, (1,2,3) and (1,3,2) are two different solutions, while (1,2*,2) and (1,2,2*) are the same solution.(* is just a symbol) Are there any ideas about this? Thanks in advance!
 A: Let's write $\displaystyle x = \prod_{i=1}^n p_i^{a_i}$, $\displaystyle y = \prod_{i=1}^n p_i^{b_i}$, $\displaystyle z = \prod_{i=1}^n p_i^{c_i}$, where the $p_i$ are the primes that are factors of at least one of $x, y, z$.
Then $\displaystyle \gcd(x,y,z) = \prod_{i=1}^n p_i^{\min(a_i,b_i,c_i)}$ and $\displaystyle \mathrm{lcm}(x,y,z) = \prod_{i=1}^n p_i^{\max(a_i,b_i,c_i)}$.
Now, suppose that $n=1$ for simplicity.  Then the number of $(x,y,z)$ with gcd $p^q$ and lcm $p^r$ is just the number of triples $(a,b,c)$ with minimum $q$ and maximum $r$.  If $r=q$ this is just $1$, while if $q<r$ this is $6(r-q-1) +3+3 = 6(r-q)$ where the first term counts the triples where all three are different and the other two terms are for triples where two of them are equal (either two equal to $p^q$ or two equal to $p^r$).  If $r<q$ the number is $0$.
So define $f(s) = 0$ if $s<0$, $f(0) = 1$, and $f(s) = 6s$ if $s>0$.
Now let $\displaystyle G = \prod_{i=1}^n p_i^{q_i}$ and $\displaystyle L = \prod_{i=1}^n p_i^{r_i}$.  Then one can show that the number of triples with gcd $G$ and lcm $L$ is $\displaystyle \prod_{i=1}^n f(r_i-q_i)$
A: Let $G=p_1^{\alpha_1}\cdot\ldots\cdot p_n^{\alpha_n}$ with $p_i$ different prime numbers and $\frac{L}{GCD(L,G)}=q_1^{\beta_1}\cdot\ldots\cdot q_n^{\beta_n}$ with $q_i$ different prime numbers.
Now $(x, y, z)$ satisfies your conditions if and olny if $x=G\cdot l$, $y= G\cdot h$, $z= G\cdot k$ where $(l,h,k)$ are all the possible factorization of $\frac{L}{GCD(L,G)}$.
Here you need some combinatorics about $\frac{L}{GCD(L,G)}=q_1^{\beta_1}\cdot\ldots\cdot q_n^{\beta_n}$
