For the natural numbers `n_1 and n_2` if `2n_1 + 3n_2 = n_1 xx n_2`, then the least possible value of the `2n_1 + 3n_2` is :

Options

A. a. 6
B. b. 12
C. c. 24
D. d. can't be determined

Answer

To solve the equation \(2n_1 + 3n_2 = n_1 \times n_2\) for natural numbers \(n_1\) and \(n_2\) and find the least possible value of \(2n_1 + 3n_2\), we can use a systematic approach.

### Step 1: Rearranging the Equation
We start with the equation:
\[ 2n_1 + 3n_2 = n_1 \times n_2 \]
We can rearrange it to:
\[ n_1 \times n_2 - 2n_1 - 3n_2 = 0 \]

### Step 2: Expressing \(n_2\) in terms of \(n_1\)
Rearranging gives us:
\[ n_1 n_2 - 2n_1 = 3n_2 \]
This can be rewritten as:
\[ n_1 n_2 - 3n_2 = 2n_1 \]
Factoring out \(n_2\):
\[ n_2(n_1 - 3) = 2n_1 \]
Thus, we can express \(n_2\) as:
\[ n_2 = \frac{2n_1}{n_1 - 3} \]

### Step 3: Finding Natural Numbers
For \(n_2\) to be a natural number, \(n_1 - 3\) must divide \(2n_1\). We will test different values of \(n_1\) starting from the smallest natural number greater than 3.

#### Testing \(n_1 = 4\):
\[ n_2 = \frac{2 \times 4}{4 - 3} = \frac{8}{1} = 8 \]
Now we calculate \(2n_1 + 3n_2\):
\[ 2n_1 + 3n_2 = 2 \times 4 + 3 \times 8 = 8 + 24 = 32 \]

#### Testing \(n_1 = 5\):
\[ n_2 = \frac{2 \times 5}{5 - 3} = \frac{10}{2} = 5 \]
Calculating \(2n_1 + 3n_2\):
\[ 2n_1 + 3n_2 = 2 \times 5 + 3 \times 5 = 10 + 15 = 25 \]

#### Testing \(n_1 = 6\):
\[ n_2 = \frac{2 \times 6}{6 - 3} = \frac{12}{3} = 4 \]
Calculating \(2n_1 + 3n_2\):
\[ 2n_1 + 3n_2 = 2 \times 6 + 3 \times 4 = 12 + 12 = 24 \]

#### Testing \(n_1 = 7\):
\[ n_2 = \frac{2 \times 7}{7 - 3} = \frac{14}{4} = 3.5 \quad (\text{not a natural number}) \]

#### Testing \(n_1 = 8\):
\[ n_2 = \frac{2 \times 8}{8 - 3} = \frac{16}{5} = 3.2 \quad (\text{not a natural number}) \]

### Conclusion
The least possible value of \(2n_1 + 3n_2\) occurs when \(n_1 = 6\) and \(n_2 = 4\), giving us:
\[ \text{Least possible value} = 24 \]