Question

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

A
36
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B
90
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C
120
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D
None of these
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Solution

## The correct option is C 12024 = 2.3.4, 2.2.6, 1.3.8, 1.2.12, 1.1.24 (as product of three positive integers) ∴ the total number of positive integral solution of xyz = 24 is equal to 3!+3!2!+3!+3!+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 ∴ 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. ∴ the number of negative integral solutions for these two cases = 9 x 2 = 18 ∴, Total number of possible integral solutions = 30 + 72 + 18 = 120

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