If `x^2 - 70x + lambda =0` have roots `alpha , beta in N`, `(lambda)/2, (lambda)/3 notin N`. Find minimum value of `lambda`

Options

A. 320
B. 325
C. 330
D. 335

Answer

To solve the problem, we need to find the minimum value of \(\lambda\) such that the quadratic equation \(x^2 - 70x + \lambda = 0\) has roots \(\alpha\) and \(\beta\) that are natural numbers, and \(\frac{\lambda}{2}\) and \(\frac{\lambda}{3}\) are not natural numbers.

### Step-by-step Solution:

1. **Understanding the Roots**:
From Vieta's formulas, we know:
\[ \alpha + \beta = 70 \]
\[ \alpha \cdot \beta = \lambda \]

2. **Expressing \(\beta\)**:
Since \(\alpha + \beta = 70\), we can express \(\beta\) in terms of \(\alpha\):
\[ \beta = 70 - \alpha \]

3. **Finding \(\lambda\)**:
Substitute \(\beta\) into the product:
\[ \lambda = \alpha \cdot (70 - \alpha) = 70\alpha - \alpha^2 \]

4. **Conditions on \(\lambda\)**:
We need \(\frac{\lambda}{2}\) and \(\frac{\lambda}{3}\) not to be natural numbers. This implies that \(\lambda\) must not be divisible by 2 or 3.

5. **Testing Values of \(\alpha\)**:
Since \(\alpha\) and \(\beta\) are natural numbers, \(\alpha\) must be at least 1 and at most 69. We will check values of \(\alpha\) starting from the smallest odd number (since \(\lambda\) must be odd to not be divisible by 2).

- **For \(\alpha = 5\)**:
\[ \beta = 70 - 5 = 65 \]
\[ \lambda = 5 \cdot 65 = 325 \]
- Check divisibility:
\[ \frac{325}{2} = 162.5 \quad (\text{not a natural number}) \]
\[ \frac{325}{3} \approx 108.33 \quad (\text{not a natural number}) \]
- Thus, \(\lambda = 325\) satisfies the conditions.

6. **Checking Larger Values of \(\alpha\)**:
- **For \(\alpha = 7\)**:
\[ \beta = 70 - 7 = 63 \]
\[ \lambda = 7 \cdot 63 = 441 \]
- Check divisibility:
\[ \frac{441}{2} = 220.5 \quad (\text{not a natural number}) \]
\[ \frac{441}{3} = 147 \quad (\text{is a natural number}) \]
- Therefore, \(\lambda = 441\) does not satisfy the conditions.

Continuing this way, we find that the next odd values of \(\alpha\) will yield larger values of \(\lambda\) which will either be divisible by 2 or 3.

7. **Conclusion**:
The minimum value of \(\lambda\) that satisfies all conditions is:
\[ \boxed{325} \]