The number of positive integral solutions of abc = 42 is:

Options

A. 17
B. 27
C. 21
D. `3! ""xx 42`

Answer

To find the number of positive integral solutions of the equation \(abc = 42\), we can follow these steps:

### Step 1: Prime Factorization of 42
First, we need to factor the number 42 into its prime factors.
\[ 42 = 2^1 \times 3^1 \times 7^1 \]

### Step 2: Using the Formula for Positive Integral Solutions
The number of positive integral solutions for the equation \(abc = n\) can be determined using the formula:
\[ \text{Number of solutions} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots (e_k + 1) \]
where \(e_i\) are the powers of the prime factors in the prime factorization of \(n\).

In our case, we have:
- For \(2\), the exponent \(e_1 = 1\)
- For \(3\), the exponent \(e_2 = 1\)
- For \(7\), the exponent \(e_3 = 1\)

### Step 3: Applying the Formula
Now we apply the formula:
\[ \text{Number of solutions} = (1 + 1)(1 + 1)(1 + 1) = 2 \times 2 \times 2 \]

### Step 4: Calculating the Result
Calculating the above expression gives:
\[ 2 \times 2 \times 2 = 8 \]

### Conclusion
Thus, the number of positive integral solutions of \(abc = 42\) is \(8\).

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