The number of the positive integral solutions `(x,y, z)` of the equation xyz = 24 is t, then the number of all possible factors of t is

Answer

To solve the problem of finding the number of positive integral solutions \((x, y, z)\) of the equation \(xyz = 24\) and then determining the number of factors of \(t\), we can follow these steps:

### Step 1: Factorization of 24
First, we need to factor the number 24 into its prime factors:
\[ 24 = 2^3 \times 3^1 \]

### Step 2: Finding the Number of Positive Integral Solutions
To find the number of positive integral solutions to the equation \(xyz = 24\), we can use the stars and bars combinatorial method. We need to distribute the exponents of the prime factors among \(x\), \(y\), and \(z\).

Let:
- \(x = 2^{a_1} \times 3^{b_1}\)
- \(y = 2^{a_2} \times 3^{b_2}\)
- \(z = 2^{a_3} \times 3^{b_3}\)

From the equation \(xyz = 24\), we have:
\[ a_1 + a_2 + a_3 = 3 \quad \text{(for the factor of 2)} \]
\[ b_1 + b_2 + b_3 = 1 \quad \text{(for the factor of 3)} \]

#### Step 2.1: Solving for \(a_1 + a_2 + a_3 = 3\)
Using the stars and bars method, the number of non-negative integer solutions to the equation \(a_1 + a_2 + a_3 = 3\) is given by:
\[ \binom{3 + 3 - 1}{3 - 1} = \binom{5}{2} = 10 \]

#### Step 2.2: Solving for \(b_1 + b_2 + b_3 = 1\)
Similarly, for the equation \(b_1 + b_2 + b_3 = 1\), the number of non-negative integer solutions is:
\[ \binom{1 + 3 - 1}{3 - 1} = \binom{3}{2} = 3 \]

### Step 3: Total Positive Integral Solutions
The total number of positive integral solutions \(t\) is the product of the solutions for \(a\) and \(b\):
\[ t = 10 \times 3 = 30 \]

### Step 4: Finding the Number of Factors of \(t\)
Next, we need to find the number of factors of \(t = 30\). First, we factor 30 into its prime factors:
\[ 30 = 2^1 \times 3^1 \times 5^1 \]

The formula for finding the number of factors from the prime factorization is:
\[ (\text{exponent of } p_1 + 1)(\text{exponent of } p_2 + 1)(\text{exponent of } p_3 + 1) \]
Applying this to our factorization:
\[ (1 + 1)(1 + 1)(1 + 1) = 2 \times 2 \times 2 = 8 \]

### Final Answer
Thus, the number of all possible factors of \(t\) is:
\[ \boxed{8} \]