To find the number of elements in the set \(\{(a,b) : a^2 + b^2 = 50, a, b \in \mathbb{Z}\}\), we will follow these steps:
### Step 1: Understand the equation
The equation \(a^2 + b^2 = 50\) implies that both \(a\) and \(b\) are integers. Since both \(a^2\) and \(b^2\) must be non-negative, we know that \(a^2\) and \(b^2\) must be less than or equal to 50.
### Step 2: Determine the possible values for \(a\)
Since \(a^2 < 50\), the maximum integer value for \(a\) can be found by calculating \(\lfloor \sqrt{50} \rfloor\). The square root of 50 is approximately 7.07, so the possible integer values for \(a\) range from \(-7\) to \(7\).
### Step 3: Calculate \(b^2\) for each integer \(a\)
We will check each integer value of \(a\) from \(-7\) to \(7\) and calculate \(b^2 = 50 - a^2\). We need to check if \(b^2\) is a perfect square.
1. **For \(a = 0\)**:
\[
b^2 = 50 - 0^2 = 50 \quad \text{(not a perfect square)}
\]
2. **For \(a = 1\)**:
\[
b^2 = 50 - 1^2 = 49 \quad (b = \pm 7)
\]
3. **For \(a = 2\)**:
\[
b^2 = 50 - 2^2 = 46 \quad \text{(not a perfect square)}
\]
4. **For \(a = 3\)**:
\[
b^2 = 50 - 3^2 = 41 \quad \text{(not a perfect square)}
\]
5. **For \(a = 4\)**:
\[
b^2 = 50 - 4^2 = 34 \quad \text{(not a perfect square)}
\]
6. **For \(a = 5\)**:
\[
b^2 = 50 - 5^2 = 25 \quad (b = \pm 5)
\]
7. **For \(a = 6\)**:
\[
b^2 = 50 - 6^2 = 14 \quad \text{(not a perfect square)}
\]
8. **For \(a = 7\)**:
\[
b^2 = 50 - 7^2 = 1 \quad (b = \pm 1)
\]
### Step 4: List the valid pairs \((a, b)\)
From the calculations above, we have the following valid pairs:
- For \(a = 1\): \((1, 7)\), \((1, -7)\)
- For \(a = -1\): \((-1, 7)\), \((-1, -7)\)
- For \(a = 5\): \((5, 5)\), \((5, -5)\)
- For \(a = -5\): \((-5, 5)\), \((-5, -5)\)
- For \(a = 7\): \((7, 1)\), \((7, -1)\)
- For \(a = -7\): \((-7, 1)\), \((-7, -1)\)
### Step 5: Count the total number of pairs
Now we can count the total number of unique pairs:
- From \(a = 1\): 2 pairs
- From \(a = -1\): 2 pairs
- From \(a = 5\): 2 pairs
- From \(a = -5\): 2 pairs
- From \(a = 7\): 2 pairs
- From \(a = -7\): 2 pairs
Adding these gives us:
\[
2 + 2 + 2 + 2 + 2 + 2 = 12
\]
### Final Answer
The total number of elements in the set is **12**.
