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 will first determine the prime factorization of 30 and then use it to find the combinations of factors that multiply to give 30.

### Step-by-Step Solution:

1. **Prime Factorization of 30**:
\[ 30 = 2^1 \times 3^1 \times 5^1 \]

2. **Finding the Total Number of Factors**:
To find the number of positive integral solutions for \(abc = 30\), we need to consider the ways to distribute the prime factors among \(a\), \(b\), and \(c\).

3. **Using the Stars and Bars Method**:
Each prime factor can be distributed among \(a\), \(b\), and \(c\). For each prime factor, we can use the stars and bars method to find the number of ways to distribute them.

- For \(2^1\): We can distribute 1 factor of 2 among \(a\), \(b\), and \(c\). The equation can be represented as:
\[ x_1 + x_2 + x_3 = 1 \]
where \(x_1\), \(x_2\), and \(x_3\) represent the number of factors of 2 in \(a\), \(b\), and \(c\) respectively. The number of non-negative integer solutions is given by:
\[ \binom{1 + 3 - 1}{3 - 1} = \binom{3}{2} = 3 \]

- For \(3^1\): Similarly, we distribute 1 factor of 3:
\[ y_1 + y_2 + y_3 = 1 \]
The number of solutions is:
\[ \binom{1 + 3 - 1}{3 - 1} = \binom{3}{2} = 3 \]

- For \(5^1\): Again, we distribute 1 factor of 5:
\[ z_1 + z_2 + z_3 = 1 \]
The number of solutions is:
\[ \binom{1 + 3 - 1}{3 - 1} = \binom{3}{2} = 3 \]

4. **Calculating the Total Number of Solutions**:
Since the distributions of the prime factors are independent, we multiply the number of solutions for each prime factor:
\[ \text{Total Solutions} = 3 \times 3 \times 3 = 27 \]

5. **Conclusion**:
Therefore, the total number of positive integral solutions of \(abc = 30\) is:
\[ \boxed{27} \]