## Question

Let *a* be a factor of 120, then the number of integral solution of *x*_{1}*x*_{2}*x*_{3} = *a* is

### Solution

320

Let *x*_{4} be such that *x*_{4} = 120/*a*. then, the number of positive integral solutions of *x*_{1}*x*_{2}*x*_{3} = *a* is same as that of number of positive integral solutions of *x*_{1}*x*_{2}*x*_{3}*x*_{4} = 120 = 2^{3} × 3 × 5.

We can assign 3 and 5 to unknown quantities in 4 × 4 ways. We can assign all 2 to one unknown in ^{4}*C*_{1} ways, to two unknowns in (^{4}*C*_{2})(2) and to three unknown in ^{4}*C*_{3} ways. Hence, the number of required solutions

= 4 × 4 × [^{4}C_{1} + (^{4}C_{2})(2) + ^{4}C_{3}]

= 4 × 4 × 20 = 320.

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