The number of elements in the set `{(a, b) : 2a^2 + 3b^2 = 35. a . b in Z}`,where `Z` is the set of all integers, is

Options

A. 2
B. 4
C. 8
D. 12

Explanation

To solve the problem of finding the number of integer pairs \((a, b)\) such that \(2a^2 + 3b^2 = 35\), we can follow these steps:

### Step-by-Step Solution:

1. **Set Up the Equation**: We start with the equation given in the problem:
\[ 2a^2 + 3b^2 = 35 \]

2. **Rearranging the Equation**: We can rearrange the equation to isolate \(3b^2\):
\[ 3b^2 = 35 - 2a^2 \]

3. **Finding Integer Solutions**: Since \(b^2\) must be a non-negative integer, \(35 - 2a^2\) must be non-negative and divisible by 3. Thus, we need:
\[ 35 - 2a^2 \geq 0 \quad \text{and} \quad 35 - 2a^2 \equiv 0 \mod 3 \]

4. **Finding Possible Values for \(a\)**:
- We can find possible integer values for \(a\) such that \(2a^2 \leq 35\):
- \(a = 0\): \(2(0^2) = 0 \Rightarrow 3b^2 = 35 \Rightarrow b^2 = \frac{35}{3}\) (not an integer)
- \(a = 1\): \(2(1^2) = 2 \Rightarrow 3b^2 = 33 \Rightarrow b^2 = 11\) (not an integer)
- \(a = 2\): \(2(2^2) = 8 \Rightarrow 3b^2 = 27 \Rightarrow b^2 = 9 \Rightarrow b = \pm 3\) (valid solutions)
- \(a = 3\): \(2(3^2) = 18 \Rightarrow 3b^2 = 17\) (not divisible by 3)
- \(a = 4\): \(2(4^2) = 32 \Rightarrow 3b^2 = 3 \Rightarrow b^2 = 1 \Rightarrow b = \pm 1\) (valid solutions)
- \(a = 5\): \(2(5^2) = 50\) (exceeds 35)

5. **Listing Valid Solutions**:
- From \(a = 2\): \(b = 3\) or \(b = -3\) gives us the pairs \((2, 3)\) and \((2, -3)\).
- From \(a = 4\): \(b = 1\) or \(b = -1\) gives us the pairs \((4, 1)\) and \((4, -1)\).

6. **Considering Negative Values of \(a\)**:
- Since \(b^2\) is the same for \(a\) and \(-a\), we can also consider negative values:
- For \(a = -2\): we get pairs \((-2, 3)\) and \((-2, -3)\).
- For \(a = -4\): we get pairs \((-4, 1)\) and \((-4, -1)\).

7. **Counting All Unique Solutions**:
- The valid pairs are:
- \((2, 3)\), \((2, -3)\)
- \((4, 1)\), \((4, -1)\)
- \((-2, 3)\), \((-2, -3)\)
- \((-4, 1)\), \((-4, -1)\)

Thus, we have a total of 8 unique pairs.

### Final Answer:
The total number of elements in the set is **8**.