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Question

The total number of integral solutions for (x,y,z) such that xyz=24 is
  1. 36
  2. 90
  3. 96
  4. 120


Solution

The correct option is D 120
Factors of 24:
24=1×2×2×2×3
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 3!2! to consider permutation. So, there are 3 ways to permute these elements.
Hence, there are 4×6+2×3=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 4×30=120 integral solutions of xyz=24

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