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

Options

A. 30
B. 90
C. 120
D. None of these

Answer

To find the total number of integral solutions for the equation \( xyz = 24 \), we will follow these steps:

### Step 1: Factor the number 24
First, we need to express 24 in terms of its prime factors.

\[ 24 = 2^3 \times 3^1 \]

### Step 2: Express variables in terms of prime factors
Next, we can express \( x, y, z \) in terms of their prime factors. Let:

\[ x = 2^{a_1} \times 3^{b_1}, \quad y = 2^{a_2} \times 3^{b_2}, \quad z = 2^{a_3} \times 3^{b_3} \]

### Step 3: Set up equations for the powers
From the equation \( xyz = 24 \), we can equate the powers of the prime factors:

1. For the power of 2:
\[ a_1 + a_2 + a_3 = 3 \]

2. For the power of 3:
\[ b_1 + b_2 + b_3 = 1 \]

### Step 4: Find non-negative integral solutions for the power of 2
Now we need to find the number of non-negative integral solutions for the equation \( a_1 + a_2 + a_3 = 3 \). This can be solved using the "stars and bars" theorem.

The number of solutions is given by:
\[ \text{Number of solutions} = \binom{n+k-1}{k-1} \]
where \( n \) is the total we want (3) and \( k \) is the number of variables (3):

\[ \text{Number of solutions} = \binom{3 + 3 - 1}{3 - 1} = \binom{5}{2} = 10 \]

### Step 5: Find non-negative integral solutions for the power of 3
Next, we find the number of non-negative integral solutions for the equation \( b_1 + b_2 + b_3 = 1 \):

\[ \text{Number of solutions} = \binom{1 + 3 - 1}{3 - 1} = \binom{3}{2} = 3 \]

### Step 6: Calculate total solutions
Finally, we multiply the number of solutions for the powers of 2 and 3 to get the total number of integral solutions for \( (x, y, z) \):

\[ \text{Total solutions} = 10 \times 3 = 30 \]

Thus, the total number of integral solutions for \( (x, y, z) \) such that \( xyz = 24 \) is \( \boxed{30} \).

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