Question

The number of all possible positive integral solutions of the equation \(xyz = 30\) is:

• A. 24
• B. 25
• C. 26
• D. 27

Hint:

Number of positive integral solutions of \(xyz = n\) for \(n = p_1^{a_1} p_2^{a_2} \cdots\) is the product of combinations with repetition.

Answer 1

The Correct Option is D

Step 1: Factorize 30
\[ 30 = 2 \times 3 \times 5 \] Step 2: Number of positive integer solutions for \(xyz = 30\)
Number of ordered positive integer solutions equals the number of ways to distribute prime factors among \(x,y,z\). Step 3: Use stars and bars method
Number of solutions: \[ (1+3-1)(1+3-1)(1+3-1) = 3 \times 3 \times 3 = 27 \]