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(C) To find the number of positive integral solutions of $xyz = 90$,we first find the prime factorization of $90$.$90 = 2^1 	imes 3^2 	imes 5^1$.Sin

Vedclass · Accepted Answer

$54$

Vedclass · Answer

(C) To find the number of positive integral solutions of $xyz = 90$,we first find the prime factorization of $90$.$90 = 2^1 	imes 3^2 	imes 5^1$.Since $x, y, z$ are positive integers,they must be of the form $x = 2^{a_1} 3^{b_1} 5^{c_1}$,$y = 2^{a_2} 3^{b_2} 5^{c_2}$,and $z = 2^{a_3} 3^{b_3} 5^{c_3}$.The product $xyz = 2^{a_1+a_2+a_3} 	imes 3^{b_1+b_2+b_3} 	imes 5^{c_1+c_2+c_3} = 2^1 	imes 3^2 	imes 5^1$.This gives us three independent equations for the exponents:$a_1 + a_2 + a_3 = 1$,where $a_i \ge 0$.The number of non-negative integer solutions is given by the stars and bars formula $\binom{n+k-1}{k-1} = \binom{1+3-1}{3-1} = \binom{3}{2} = 3$.$b_1 + b_2 + b_3 = 2$,where $b_i \ge 0$.The number of solutions is $\binom{2+3-1}{3-1} = \binom{4}{2} = 6$.$c_1 + c_2 + c_3 = 1$,where $c_i \ge 0$.The number of solutions is $\binom{1+3-1}{3-1} = \binom{3}{2} = 3$.The total number of solutions is the product of the number of solutions for each prime factor: $3 	imes 6 	imes 3 = 54$.

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

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