Question
The total number of integral solutions for $$\left( {x,y,z} \right)$$ such that $$xyz = 24$$ is
A.
36
B.
90
C.
120
D.
None of these
Answer :
120
Solution :
$$24 = 2 \cdot 3 \cdot 4,2 \cdot 2 \cdot 6,1 \cdot 6 \cdot 4,1 \cdot 3 \cdot 8,1 \cdot 2 \cdot 12,1 \cdot 1 \cdot 24$$
(as product of three positive integers)
∴ the total number of positive integral solutions of $$xyz = 24$$ is equal to $$3!\, + \frac{{3!}}{{2!}} + 3!\, + 3!\, + 3!\, + \frac{{3!}}{{2!}},{\text{i}}{\text{.e}}{\text{., }}30.$$
Any two of the factors in each factorization may be negative.
∴ the number of ways to associate negative sign in each case is $$^3{C_2},$$ i.e., 3.
∴ the total number of integral solutions $$ = 30 + 3 \times 30 = 120.$$