To find the number of integer pairs \((a, b)\) such that \(a^2 + b^2 = 50\), we will follow these steps:
### Step 1: Understand the equation
We start with the equation:
\[
a^2 + b^2 = 50
\]
This means we need to find integer values for \(a\) and \(b\) such that their squares add up to 50.
### Step 2: Determine the range for \(a\)
Since \(a^2\) must be less than or equal to 50, we can find the maximum possible value for \(a\):
\[
a^2 \leq 50 \implies |a| \leq \sqrt{50} \approx 7.07
\]
Thus, \(a\) can take integer values from \(-7\) to \(7\).
### Step 3: Check possible values of \(a\)
We will check each integer value of \(a\) from \(-7\) to \(7\) and see if \(b^2\) is a perfect square.
- **For \(a = 7\)**:
\[
b^2 = 50 - 7^2 = 50 - 49 = 1 \implies b = \pm 1
\]
Solutions: \((7, 1)\), \((7, -1)\)
- **For \(a = 6\)**:
\[
b^2 = 50 - 6^2 = 50 - 36 = 14 \quad \text{(not a perfect square)}
\]
- **For \(a = 5\)**:
\[
b^2 = 50 - 5^2 = 50 - 25 = 25 \implies b = \pm 5
\]
Solutions: \((5, 5)\), \((5, -5)\)
- **For \(a = 4\)**:
\[
b^2 = 50 - 4^2 = 50 - 16 = 34 \quad \text{(not a perfect square)}
\]
- **For \(a = 3\)**:
\[
b^2 = 50 - 3^2 = 50 - 9 = 41 \quad \text{(not a perfect square)}
\]
- **For \(a = 2\)**:
\[
b^2 = 50 - 2^2 = 50 - 4 = 46 \quad \text{(not a perfect square)}
\]
- **For \(a = 1\)**:
\[
b^2 = 50 - 1^2 = 50 - 1 = 49 \implies b = \pm 7
\]
Solutions: \((1, 7)\), \((1, -7)\)
- **For \(a = 0\)**:
\[
b^2 = 50 - 0^2 = 50 \quad \text{(not a perfect square)}
\]
- **For \(a = -1\)**:
\[
b^2 = 50 - (-1)^2 = 50 - 1 = 49 \implies b = \pm 7
\]
Solutions: \((-1, 7)\), \((-1, -7)\)
- **For \(a = -2\)**:
\[
b^2 = 50 - (-2)^2 = 50 - 4 = 46 \quad \text{(not a perfect square)}
\]
- **For \(a = -3\)**:
\[
b^2 = 50 - (-3)^2 = 50 - 9 = 41 \quad \text{(not a perfect square)}
\]
- **For \(a = -4\)**:
\[
b^2 = 50 - (-4)^2 = 50 - 16 = 34 \quad \text{(not a perfect square)}
\]
- **For \(a = -5\)**:
\[
b^2 = 50 - (-5)^2 = 50 - 25 = 25 \implies b = \pm 5
\]
Solutions: \((-5, 5)\), \((-5, -5)\)
- **For \(a = -6\)**:
\[
b^2 = 50 - (-6)^2 = 50 - 36 = 14 \quad \text{(not a perfect square)}
\]
- **For \(a = -7\)**:
\[
b^2 = 50 - (-7)^2 = 50 - 49 = 1 \implies b = \pm 1
\]
Solutions: \((-7, 1)\), \((-7, -1)\)
### Step 4: Count all valid pairs
Now we list all the valid pairs:
1. \((7, 1)\)
2. \((7, -1)\)
3. \((5, 5)\)
4. \((5, -5)\)
5. \((1, 7)\)
6. \((1, -7)\)
7. \((-1, 7)\)
8. \((-1, -7)\)
9. \((-5, 5)\)
10. \((-5, -5)\)
11. \((-7, 1)\)
12. \((-7, -1)\)
Thus, there are a total of **12 pairs**.
### Final Answer
The number of elements in the set is **12**.
