The number of positive integral solutions of | vedclass

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(C) The prime factorization of $90$ is $90 = 2^1 	imes 3^2 	imes 5^1$. Let $x = 2^{x_1} 3^{y_1} 5^{z_1}$,$y = 2^{x_2} 3^{y_2} 5^{z_2}$,and $z = 2^{x

Vedclass · Accepted Answer

$54$

Vedclass · Answer

(C) The prime factorization of $90$ is $90 = 2^1 	imes 3^2 	imes 5^1$. Let $x = 2^{x_1} 3^{y_1} 5^{z_1}$,$y = 2^{x_2} 3^{y_2} 5^{z_2}$,and $z = 2^{x_3} 3^{y_3} 5^{z_3}$,where $x_i, y_i, z_i \ge 0$. Since $xyz = 90$,we have: $x_1 + x_2 + x_3 = 1$ $y_1 + y_2 + y_3 = 2$ $z_1 + z_2 + z_3 = 1$ Using the stars and bars formula $\binom{n+k-1}{k-1}$,the number of solutions for each variable is: For $x$: $\binom{1+3-1}{3-1} = \binom{3}{2} = 3$ For $y$: $\binom{2+3-1}{3-1} = \binom{4}{2} = 6$ For $z$: $\binom{1+3-1}{3-1} = \binom{3}{2} = 3$ Total number of solutions = $3 	imes 6 	imes 3 = 54$.

The number of positive integral solutions of the equation $xyz = 90$ is equal to :-

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