The total number of integral solutions for th | vedclass

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Question

(B) First,find the number of positive integral solutions for $xyz = 24$.$24 = 2^3 	imes 3^1$.The number of positive integral solutions is given by th

Vedclass · Accepted Answer

$120$

Vedclass · Answer

(B) First,find the number of positive integral solutions for $xyz = 24$.$24 = 2^3 	imes 3^1$.The number of positive integral solutions is given by the product of the number of ways to distribute the prime factors among $x, y, z$.For $2^3$: Using stars and bars,the number of ways is $\binom{3+3-1}{3-1} = \binom{5}{2} = 10$.For $3^1$: Using stars and bars,the number of ways is $\binom{1+3-1}{3-1} = \binom{3}{2} = 3$.Total positive integral solutions $= 10 	imes 3 = 30$.Since the product is positive $(24 > 0)$,the possible sign combinations for $(x, y, z)$ are $(+, +, +)$,$(+, -, -)$,$(-, +, -)$,and $(-, -, +)$.Each case yields $30$ solutions.Total integral solutions $= 30 	imes 4 = 120$.

The total number of integral solutions for the equation $xyz = 24$ is:

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