To find the number of integral solutions (x, y, z) of the equation \(xyz = 18\), we can follow these steps:
### Step 1: Factorize the Number
First, we need to factorize the number 18 into its prime factors:
\[
18 = 2^1 \times 3^2
\]
### Step 2: Determine Positive Integral Solutions
Next, we find the positive integral solutions for the equation \(xyz = 18\). We can list the combinations of positive integers (x, y, z) that multiply to give 18:
1. \( (1, 1, 18) \)
2. \( (1, 2, 9) \)
3. \( (1, 3, 6) \)
4. \( (2, 3, 3) \)
### Step 3: Count Arrangements for Each Combination
Now, we need to count the arrangements of these combinations since (x, y, z) can be in any order:
1. For \( (1, 1, 18) \):
- Arrangements = \( \frac{3!}{2!} = 3 \) (because 1 is repeated)
2. For \( (1, 2, 9) \):
- Arrangements = \( 3! = 6 \) (all different)
3. For \( (1, 3, 6) \):
- Arrangements = \( 3! = 6 \) (all different)
4. For \( (2, 3, 3) \):
- Arrangements = \( \frac{3!}{2!} = 3 \) (because 3 is repeated)
### Step 4: Total Positive Integral Solutions
Now, we sum the arrangements:
\[
3 + 6 + 6 + 3 = 18
\]
### Step 5: Consider Negative Solutions
Next, we consider the signs of the integers. The product \(xyz = 18\) can be achieved with different combinations of positive and negative integers. The possible sign combinations are:
1. All positive: \( (+, +, +) \)
2. One positive and two negatives: \( (+, -, -) \)
3. Two positives and one negative: \( (-, +, +) \)
The only valid sign combinations for \(xyz = 18\) are:
- All positive (1 way)
- One positive and two negatives (3 ways)
- Two positives and one negative (3 ways)
Thus, we have a total of 4 sign combinations.
### Step 6: Total Integral Solutions
Now we multiply the total positive solutions by the number of sign combinations:
\[
t = 18 \times 4 = 72
\]
### Step 7: Calculate \( \frac{t}{8} \)
Finally, we calculate:
\[
\frac{t}{8} = \frac{72}{8} = 9
\]
### Final Answer
Thus, the value of \( \frac{t}{8} \) is:
\[
\boxed{9}
\]
