Let `y` be an element of the set `A={1,2,3,4,5,6,10,15,30}` 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

Options

A. 27
B. 64
C. 81
D. 256

Answer

B

Explanation

Number of solutions of the given equations is the same as the number of solutions of the equation
`x_(1)x_(2)x_(3)x_(4)=30=2xx3xx5`
Here, `x_(4)` is infact dummy variable.
If ` x_(1)x_(2)x_(3)=15`, then `x_(4)=2` and `iffx_(1)x_(2)x_(3)=5`, then `x_(4)=6`, etc.
thus, `x_(1)x_(2)x_(3)x_(4)=2xx3xx5`
Each of 2,3, and 5 will be facctor of exactly one of `x_(1),x_(2),x_(3), x_(4)` in 4 ways.
`therefore`Required number`=4^(3)=64`