Question

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

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

Hint:

If \[ xyz=n \] and \[ n=p_1^{a_1}p_2^{a_2}\cdots, \] then distribute each prime exponent among the variables using stars and bars, and multiply the resulting counts.

Answer 1

The Correct Option is B

Step 1: Prime factorization of \(24\).
\[ 24=2^3\cdot 3^1. \] We need the number of ordered positive integral triples \[ (x,y,z) \] such that \[ xyz=2^3\cdot 3. \]

Step 2: Distribute the powers of \(2\).
Let the exponents of \(2\) in \(x,y,z\) be \[ a,b,c. \] Then \[ a+b+c=3. \] The number of non-negative integral solutions is \[ \binom{3+3-1}{3-1} = \binom{5}{2} = 10. \]

Step 3: Distribute the powers of \(3\).
Let the exponents of \(3\) in \(x,y,z\) be \[ p,q,r. \] Then \[ p+q+r=1. \] The number of non-negative integral solutions is \[ \binom{1+3-1}{3-1} = \binom{3}{2} = 3. \]

Step 4: Apply the multiplication principle.
The distributions of powers of \(2\) and \(3\) are independent. Therefore, total ordered triples are \[ 10\times 3 = 30. \]

Step 5: Final conclusion.
Hence, the total number of positive integral solutions is \[ \boxed{30} \]