I tried to solve this problem but I was not able to. Can someone please tell me the way to solve this problem in simple ways? I have looked up for the solution to this problem on the web but I wasn't able to get a well explained answer. I am just looking for unordered pairs of solutions.
One more question related to this problem. So in this problem we have been asked to find number of pairs of $(a,b)$ such that $LCM(a,b)=200=2^3 \times 5^2$ but is there a generalized way to solve for triplets, quadruplets etc.too? Like how many triplets of $(a,b,c)$ such that $LCM(a,b,c)=200=2^3 \times 5^2$ or how many quadruplets of $(a,b,c,d)$ such that $LCM(a,b,c,d)=200=2^3 \times 5^2$
Hint, consider an ordered pair $(X, Y)$. Write \begin{align} X &= P_1^{a_1}P_2^{a_2}P_3^{a_3}\ldots P_n^{a_n} \\ Y &= P_1^{b_1}P_2^{b_2}P_3^{b_3}\ldots P_n^{b_n} \end{align} Here, $P_1, P_2,\ldots P_n$ are prime factors of $X$ and $Y$.
Now $$\text{LCM}(X, Y) = P_1^{max(a_1, b_1)}P_2^{max(a_2, b_2)}\ldots P_n^{max(a_n, b_n)}$$
Therefore $$\text{LCM}(X, Y) = N = P_1^{m_1}P_2^{m_2}P_3^{m_3}\ldots P_n^{mn}$$
Where $m_i=\text{max}(a_i,b_i)$. Therefore, total number of ordered pairs \begin{align} (X, Y) &= ((m_1 + 1)^2 – m_1^2))((m_2 + 1)^2 – m_2^2)\ldots((m_n + 1)^2 – m_n^2)\\ &= (2m_1+1) (2m_2+1)(2m_3+1)\ldots(2m_n+1) \end{align}