The total number of positive integral solutions
Question:

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

  1. 36

  2. 24

  3. 45

  4. 30


Correct Option: , 4

Solution:

$x y z=2^{3} \times 3^{1}$

Let $x=2^{\alpha_{1}} \times 3^{\beta_{1}}$

$y=2^{\alpha_{2}} \times 3^{\beta_{2}}$

$\mathrm{z}=2^{\alpha_{1}} \times 3^{\beta_{2}}$

Now $\alpha_{1}+\alpha_{2}+\alpha_{3}=3$.

No. of non-negative intergal sol $={ }^{5} \mathrm{C}_{2}=10$

$\& \beta_{1}+\beta_{2}+\beta_{3}=1$

No. of non-negative intergal soln $={ }^{3} \mathrm{C}_{2}=3$ Total ways $=10 \times 3=30$.

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