Question

The total number of positive integral solutions \((x, y, z)\) of \( xyz = 24 \) is:

• A. \( 24 \)
• B. \( 30 \)
• C. \( 36 \)
• D. \( 32 \)

Hint:

Positive Integer Solutions of Products}
Use prime factorization of RHS
Distribute powers using stars and bars method
Total ways = product of combinations per prime

Answer 1

The Correct Option is B

We are given the equation: \[ xyz = 24, \quad x, y, z \in \mathbb{Z}^+. \] Factorize \( 24 = 2^3 \cdot 3 \). We must distribute 3 powers of 2 and 1 power of 3 among the three variables. Each distribution of powers among \( x, y, z \) corresponds to a combination of: \[ (a_1 + a_2 + a_3 = 3), \quad (b_1 + b_2 + b_3 = 1) \] Number of positive integer solutions: \[ \text{Using stars and bars:} \quad \binom{3 + 3 - 1}{2} \cdot \binom{1 + 3 - 1}{2} = \binom{5}{2} \cdot \binom{3}{2} = 10 \cdot 3 = 30 \]