Let y be an element of the set $\mathrm{A}=\{1,2,3,5,6,10,15,30\}\text{ and }{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3$ be integers such that ${\mathrm{x}}_1{\mathrm{x}}_2{\mathrm{x}}_3=\mathrm{y}$ then the number of integral solutions of ${\mathrm{x}}_1{\mathrm{x}}_2{\mathrm{x}}_3=\mathrm{y}, \mathrm{is}:$

$$\begin{gathered} Number of solutions of the given equation is same as the number of solutions of the equation x_1{x}_2{x}_3{x}_4=30=2\times 3\times 5 \\ Here x_4 is there because if x_1{x}_2{x}_3=15 then x_4=2 and if x_1{x}_2{x}_3=5 then x_4=6 etc. x_4 is in fact a dummy variable. \\ Each of 2,3 and 5 will be a factor of exactly one of x_1,x_2,x_3,x_4 in 4 ways. \\ So required number is 4^3 =64 \end{gathered}$$