The total number of positive integral solutio | vedclass

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(D) The prime factorization of $24$ is $2^{3} 	imes 3^{1}$.Let $x = 2^{\alpha_{1}} 	imes 3^{\beta_{1}}$,$y = 2^{\alpha_{2}} 	imes 3^{\beta_{2}}$,an

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$30$

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(D) The prime factorization of $24$ is $2^{3} 	imes 3^{1}$.Let $x = 2^{\alpha_{1}} 	imes 3^{\beta_{1}}$,$y = 2^{\alpha_{2}} 	imes 3^{\beta_{2}}$,and $z = 2^{\alpha_{3}} 	imes 3^{\beta_{3}}$,where $\alpha_{i}, \beta_{i} \ge 0$.Since $xyz = 2^{3} 	imes 3^{1}$,we have $\alpha_{1} + \alpha_{2} + \alpha_{3} = 3$ and $\beta_{1} + \beta_{2} + \beta_{3} = 1$.The number of non-negative integer solutions for $\alpha_{1} + \alpha_{2} + \alpha_{3} = 3$ is given by the stars and bars formula $\binom{n+k-1}{k-1} = \binom{3+3-1}{3-1} = \binom{5}{2} = 10$.The number of non-negative integer solutions for $\beta_{1} + \beta_{2} + \beta_{3} = 1$ is $\binom{1+3-1}{3-1} = \binom{3}{2} = 3$.Therefore,the total number of positive integral solutions is $10 	imes 3 = 30$.

The total number of positive integral solutions $(x, y, z)$ such that $xyz = 24$ is

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