Number of positive integral solutions of the equation $x y z = 90$ is

120

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54

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108

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60

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Solution

The correct option is B $54$
As $x y z = 90 \Rightarrow x y z = 2^{1} \times 3^{2} \times 5^{1}$ $Let x = 2^{a_{1}} \times 3^{b_{1}} \times 5^{c_{1}} y = 2^{a_{2}} \times 3^{b_{2}} \times 5^{c_{2}} z = 2^{a_{3}} \times 3^{b_{3}} \times 5^{c_{3}} So, a_{1} + a_{2} + a_{3} = 1, Total Solutions will be =^{1 + 3 - 1} C_{3 - 1} = 3 b_{1} + b_{2} + b_{3} = 2, Total Solutions will be =^{2 + 3 - 1} C_{3 - 1} = 6 c_{1} + c_{2} + c_{3} = 1 Total Solutions will be =^{1 + 3 - 1} C_{3 - 1} = 3 ∴ Total number of solutions are = 3 \times 6 \times 3 = 54.$

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