The total number of positive integral solutions of abc=30, is

Options

A. 30
B. 27
C. 8
D. none of these

Explanation

To find the total number of positive integral solutions of the equation \(abc = 30\), we can follow these steps:

### Step 1: Prime Factorization of 30
First, we need to perform the prime factorization of 30:
\[ 30 = 2^1 \times 3^1 \times 5^1 \]

### Step 2: Distributing the Prime Factors
We can express \(a\), \(b\), and \(c\) in terms of the prime factors:
\[ a = 2^{x_1} \times 3^{y_1} \times 5^{z_1}, \quad b = 2^{x_2} \times 3^{y_2} \times 5^{z_2}, \quad c = 2^{x_3} \times 3^{y_3} \times 5^{z_3} \]
where \(x_1 + x_2 + x_3 = 1\), \(y_1 + y_2 + y_3 = 1\), and \(z_1 + z_2 + z_3 = 1\).

### Step 3: Finding Non-negative Integral Solutions
We need to find the non-negative integral solutions for each of these equations. The number of non-negative integral solutions of the equation \(x_1 + x_2 + x_3 = n\) is given by the formula:
\[ \binom{n + k - 1}{k - 1} \]
where \(n\) is the total we want to achieve, and \(k\) is the number of variables.

For our case:
- For \(x_1 + x_2 + x_3 = 1\):
\[ \text{Number of solutions} = \binom{1 + 3 - 1}{3 - 1} = \binom{3}{2} = 3 \]

- For \(y_1 + y_2 + y_3 = 1\):
\[ \text{Number of solutions} = \binom{1 + 3 - 1}{3 - 1} = \binom{3}{2} = 3 \]

- For \(z_1 + z_2 + z_3 = 1\):
\[ \text{Number of solutions} = \binom{1 + 3 - 1}{3 - 1} = \binom{3}{2} = 3 \]

### Step 4: Total Solutions
Now, we multiply the number of solutions for each prime factor:
\[ \text{Total solutions} = 3 \times 3 \times 3 = 27 \]

### Conclusion
Thus, the total number of positive integral solutions of \(abc = 30\) is:
\[ \boxed{27} \]