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Let y be an element of the set A={1,2,3,...

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

A

27

B

64

C

81

D

256

Text Solution

Verified by Experts

The correct Answer is:
B
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`
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