Question

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

• A. 36
• B. 90
• C. 96
• D. 120

Answer 1

The correct option is D 120

Factors of 24:
$24=1\times 2\times 2\times 2\times 3$
Combinaitons of $(x,y,z)$:
(1, 1 ,24), (1, 2, 12), (1, 3, 8) (1, 4, 6) (2, 3, 4) and (2, 2, 6)
So, there are 6 solutions.
(i) 4 of these triplets have distinct elements, we multiply these by $3!$ to consider permutation. So, there are 6 ways to permute these elements.
(ii) (1, 1, 24) and (2, 2, 6) have two dinstict elements, we muliply these by $\frac{3!}{2!}$ to consider permutation. So, there are 3 ways to permute these elements.
Hence, there are $4\times 6+2\times 3=30$ positive/negative integer solutions.
We must also take into account sign possibilities to get all integer solutions.
To get a positive number, the signs can be $(+++),(+--),(-+-),(--+)$
Multiplying 4 results in $4\times 30=120$ integral solutions of $xyz=24$.