If the number of integral solutions (x, y, z) of the equation xyz = 18 is t, then the value of `(t)/(8)` is

Answer

To find the number of integral solutions \((x, y, z)\) of the equation \(xyz = 18\), we will follow these steps:

### Step 1: Factor the number 18
First, we need to factor the number 18 into its prime factors:
\[ 18 = 2^1 \times 3^2 \]

### Step 2: Determine the total number of factors
To find the number of integral solutions, we will consider both positive and negative factors. The total number of factors of a number \(n = p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k}\) is given by:
\[ (e_1 + 1)(e_2 + 1) \ldots (e_k + 1) \]
For \(18 = 2^1 \times 3^2\):
- \(e_1 = 1\) (for the prime factor 2)
- \(e_2 = 2\) (for the prime factor 3)

Thus, the total number of factors is:
\[ (1 + 1)(2 + 1) = 2 \times 3 = 6 \]

### Step 3: List the positive factors
The positive factors of 18 are:
\[ 1, 2, 3, 6, 9, 18 \]

### Step 4: Count the combinations of factors
We need to find all combinations of these factors that multiply to give 18. We can consider the following combinations:
1. \( (1, 1, 18) \)
2. \( (1, 2, 9) \)
3. \( (1, 3, 6) \)
4. \( (2, 3, 3) \)

### Step 5: Calculate the arrangements for each combination
Now we will calculate the number of arrangements for each combination:
1. **For \( (1, 1, 18) \)**: The arrangements are:
\[ \frac{3!}{2!} = 3 \quad \text{(since 1 is repeated)} \]
2. **For \( (1, 2, 9) \)**: The arrangements are:
\[ 3! = 6 \]
3. **For \( (1, 3, 6) \)**: The arrangements are:
\[ 3! = 6 \]
4. **For \( (2, 3, 3) \)**: The arrangements are:
\[ \frac{3!}{2!} = 3 \quad \text{(since 3 is repeated)} \]

### Step 6: Sum the arrangements
Now, we sum all the arrangements:
\[ 3 + 6 + 6 + 3 = 18 \]

### Step 7: Consider negative solutions
Each positive solution can also have negative counterparts. Since \(xyz = 18\), we can have:
- All positive
- One negative and two positives (3 combinations)
- Two negatives and one positive (3 combinations)

Thus, the total number of integral solutions is:
\[ 1 + 3 + 3 = 7 \quad \text{(for each positive solution)} \]
So the total number of integral solutions is:
\[ 18 \times 2^3 = 18 \times 8 = 144 \]

### Step 8: Calculate \( \frac{t}{8} \)
Finally, we need to find \( \frac{t}{8} \):
\[ \frac{144}{8} = 18 \]

Thus, the final answer is:
\[ \boxed{18} \]