Question

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

• A. 36
• B. 90
• C. 120
• D. None of these

Answer 1

The correct option is C 120

24 = 2.3.4, 2.2.6, 1.3.8, 1.2.12, 1.1.24 (as product of three positive integers)
$\therefore$ the total number of positive integral solution of xyz = 24 is equal to $3!+\frac{3!}{2!}+3!+3!+\frac{3!}{2!}$, i.e., 30
Now for triplets/sets of 1.3.8, 2.3.4, 1.4.6, 1.2.12, any two of the factors in each factorization may be negative
$\therefore$ the number of negative integral solutions for such cases = 4 x 3C2 x 3! = 72
For triplets/sets 2.2.6 and 1.1.24, each will have 9 negative integral solutions.
$\therefore$ the number of negative integral solutions for these two cases = 9 x 2 = 18
$\therefore$, Total number of possible integral solutions = 30 + 72 + 18 = 120