Let $y$ be the element of the set $A=\{1,2,3,5,6,10,15,30\}$ and $x_1,x_2,x_3$ be integers  such that $x_1{x}_2{x}_3=y,$ the number of positive integral solutions of $y$ are

Number of solutions of the given equation is the 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 infact a dummy variable. 

If $x_1{x}_2{x}_3=15,then{x}_4=2$ and if $x_1{x}_2{x}_3=5,then{x}_4=6,etc$ Thus, $x_1{x}_2{x}_3{x}_4=2\times 3\times 5.$   Each  of the 2,3 and 5 will be factor of exactly one of $x_1,x_2,x_3,x_4$ in 4 ways.

$\therefore \mathrm{Re}quirednumber=4^3=64$