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 all possible factors of \(t\), we can follow these steps:

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

### Step 2: Use the formula for the number of positive integral solutions
The number of positive integral solutions for the equation \(xyz = n\) (where \(n\) is expressed in its prime factorization \(p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k}\)) can be given by the formula:
\[ (e_1 + 1)(e_2 + 1) \ldots (e_k + 1) \]
For \(24 = 2^3 \times 3^1\), we have:
- \(e_1 = 3\) (for prime \(2\))
- \(e_2 = 1\) (for prime \(3\))

Thus, the number of positive integral solutions \(t\) is:
\[ t = (3 + 1)(1 + 1) = 4 \times 2 = 8 \]

### Step 3: Find the number of factors of \(t\)
Next, we need to find the number of factors of \(t = 8\). First, we factor \(8\):
\[ 8 = 2^3 \]
Using the formula for the number of factors, which is given by \((e_1 + 1)(e_2 + 1) \ldots (e_k + 1)\), we find:
- \(e_1 = 3\) (for prime \(2\))

Thus, the number of factors of \(8\) is:
\[ (3 + 1) = 4 \]

### Final Answer
The number of all possible factors of \(t\) is \(4\).