Question

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

• A. $64$
• B. $27$
• C. $81$
• D. None of these

Answer 1

Answer: (A)

$64$

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 $4^3=64$