The number of elements in the set
`{(a,b):2a^(2)+3b^(2)=83,a,binZ}`, where Z is the set of all integers, is

Options

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

Answer

To solve the problem of finding the number of elements in the set \(\{(a,b): 2a^2 + 3b^2 = 83, a, b \in \mathbb{Z}\}\), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ 2a^2 + 3b^2 = 83 \]
We can rearrange this to express \(3b^2\):
\[ 3b^2 = 83 - 2a^2 \]

### Step 2: Finding the Range for \(b^2\)
Since \(b^2\) must be non-negative, we require:
\[ 83 - 2a^2 \geq 0 \implies 2a^2 \leq 83 \implies a^2 \leq \frac{83}{2} = 41.5 \]
This means \(a^2\) can take integer values from \(0\) to \(41\). Thus, the possible integer values for \(a\) are:
\[ a = -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 \]
This gives us a total of \(13\) possible values for \(a\).

### Step 3: Finding Corresponding Values for \(b\)
Next, we need to check for each integer value of \(a\) whether \(b^2\) results in a perfect square. We will calculate \(b^2\) for each valid \(a\):

1. **For \(a = 0\)**:
\[ 3b^2 = 83 \implies b^2 = \frac{83}{3} \quad \text{(not an integer)} \]

2. **For \(a = \pm 1\)**:
\[ 3b^2 = 83 - 2(1^2) = 81 \implies b^2 = 27 \quad (b = \pm 3\sqrt{3} \text{ (not an integer)}) \]

3. **For \(a = \pm 2\)**:
\[ 3b^2 = 83 - 2(2^2) = 75 \implies b^2 = 25 \quad (b = \pm 5) \]

4. **For \(a = \pm 3\)**:
\[ 3b^2 = 83 - 2(3^2) = 61 \implies b^2 = \frac{61}{3} \quad \text{(not an integer)} \]

5. **For \(a = \pm 4\)**:
\[ 3b^2 = 83 - 2(4^2) = 51 \implies b^2 = 17 \quad (b = \pm \sqrt{17} \text{ (not an integer)}) \]

6. **For \(a = \pm 5\)**:
\[ 3b^2 = 83 - 2(5^2) = 33 \implies b^2 = 11 \quad (b = \pm \sqrt{11} \text{ (not an integer)}) \]

7. **For \(a = \pm 6\)**:
\[ 3b^2 = 83 - 2(6^2) = 11 \implies b^2 = \frac{11}{3} \quad \text{(not an integer)} \]

### Step 4: Valid Combinations
From our calculations, we found valid pairs only for \(a = \pm 2\) and \(b = \pm 5\):
- \((2, 5)\), \((2, -5)\), \((-2, 5)\), \((-2, -5)\)

### Conclusion
Thus, the total number of integer pairs \((a, b)\) that satisfy the equation is \(4\).

### Final Answer
The number of elements in the set is \(4\).