The number of all possible positive integral | vedclass

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(D) To find the number of positive integral solutions of $xyz=30$,we first find the prime factorization of $30$: $30 = 2^1 	imes 3^1 	imes 5^1$. Let

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$27$

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(D) To find the number of positive integral solutions of $xyz=30$,we first find the prime factorization of $30$: $30 = 2^1 	imes 3^1 	imes 5^1$. Let $x = 2^{a_1} 3^{b_1} 5^{c_1}$,$y = 2^{a_2} 3^{b_2} 5^{c_2}$,and $z = 2^{a_3} 3^{b_3} 5^{c_3}$,where $a_i, b_i, c_i \ge 0$. Then $a_1+a_2+a_3 = 1$,$b_1+b_2+b_3 = 1$,and $c_1+c_2+c_3 = 1$. The number of non-negative integral solutions for each equation of the form $x_1+x_2+x_3 = n$ is given by $\binom{n+3-1}{3-1} = \binom{n+2}{2}$. For $n=1$,the number of solutions is $\binom{1+2}{2} = \binom{3}{2} = 3$. Since there are three independent variables $(a, b, c)$,the total number of solutions is $3 	imes 3 	imes 3 = 27$.

The number of all possible positive integral solutions of the equation $xyz=30$ is

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