To solve the question, we need to analyze the given equations and find the required values step by step.
### Step-by-step Solution:
1. **Understanding the Equation**:
We are given the equation \(2a^2 + 3b^2 = 35\), where \(a\) and \(b\) are integers (denoted by \(Z\)).
2. **Rearranging the Equation**:
We can rearrange the equation to find possible values of \(a\) and \(b\):
\[
3b^2 = 35 - 2a^2
\]
This implies that \(35 - 2a^2\) must be non-negative and divisible by 3.
3. **Finding Possible Values for \(a\)**:
We can find values of \(a\) such that \(2a^2 \leq 35\):
\[
a^2 \leq \frac{35}{2} \approx 17.5
\]
Thus, \(a\) can take integer values from \(-4\) to \(4\) (since \(4^2 = 16\) and \(5^2 = 25\) exceeds \(17.5\)).
4. **Calculating Corresponding \(b\) Values**:
We will substitute integer values of \(a\) from \(-4\) to \(4\) into the equation and check for valid integer \(b\) values:
- For \(a = -4\):
\[
2(-4)^2 + 3b^2 = 35 \implies 32 + 3b^2 = 35 \implies 3b^2 = 3 \implies b^2 = 1 \implies b = \pm 1
\]
- For \(a = -3\):
\[
2(-3)^2 + 3b^2 = 35 \implies 18 + 3b^2 = 35 \implies 3b^2 = 17 \implies b^2 \text{ not an integer}
\]
- For \(a = -2\):
\[
2(-2)^2 + 3b^2 = 35 \implies 8 + 3b^2 = 35 \implies 3b^2 = 27 \implies b^2 = 9 \implies b = \pm 3
\]
- For \(a = -1\):
\[
2(-1)^2 + 3b^2 = 35 \implies 2 + 3b^2 = 35 \implies 3b^2 = 33 \implies b^2 \text{ not an integer}
\]
- For \(a = 0\):
\[
2(0)^2 + 3b^2 = 35 \implies 3b^2 = 35 \implies b^2 \text{ not an integer}
\]
- For \(a = 1\):
\[
2(1)^2 + 3b^2 = 35 \implies 2 + 3b^2 = 35 \implies 3b^2 = 33 \implies b^2 \text{ not an integer}
\]
- For \(a = 2\):
\[
2(2)^2 + 3b^2 = 35 \implies 8 + 3b^2 = 35 \implies 3b^2 = 27 \implies b^2 = 9 \implies b = \pm 3
\]
- For \(a = 3\):
\[
2(3)^2 + 3b^2 = 35 \implies 18 + 3b^2 = 35 \implies 3b^2 = 17 \implies b^2 \text{ not an integer}
\]
- For \(a = 4\):
\[
2(4)^2 + 3b^2 = 35 \implies 32 + 3b^2 = 35 \implies 3b^2 = 3 \implies b^2 = 1 \implies b = \pm 1
\]
5. **Listing All Valid \((a, b)\) Pairs**:
From the calculations, we have the following valid pairs:
- For \(a = -4\): \((-4, 1)\), \((-4, -1)\)
- For \(a = -2\): \((-2, 3)\), \((-2, -3)\)
- For \(a = 2\): \((2, 3)\), \((2, -3)\)
- For \(a = 4\): \((4, 1)\), \((4, -1)\)
6. **Counting the Solutions**:
The valid pairs are:
- \((-4, 1)\)
- \((-4, -1)\)
- \((-2, 3)\)
- \((-2, -3)\)
- \((2, 3)\)
- \((2, -3)\)
- \((4, 1)\)
- \((4, -1)\)
This gives us a total of 8 valid pairs.
### Final Answer:
Thus, the total number of integer solutions \((a, b)\) is **8**.
