Which of the following nuclides has the magic number of both protons and neutrons?

Options

A. `Sn_50^115`
B. `Pb_82^206`
C. `Pb_82^208`
D. `Sn_50^118`

Answer

To determine which nuclide has a magic number of both protons and neutrons, we need to identify the magic numbers for protons and neutrons and then check the given options.

### Step-by-Step Solution:

1. **Identify Magic Numbers**:
- Magic numbers for protons: 2, 8, 20, 28, 50, 82
- Magic numbers for neutrons: 2, 8, 20, 28, 50, 82, 126

2. **Analyze Each Option**:
- **Option 1: Sn-115**
- Protons (Z) = 50 (which is a magic number)
- Neutrons (N) = 115 - 50 = 65 (not a magic number)
- Conclusion: Does not satisfy both conditions.

- **Option 2: Pb-206**
- Protons (Z) = 82 (which is a magic number)
- Neutrons (N) = 206 - 82 = 124 (not a magic number)
- Conclusion: Does not satisfy both conditions.

- **Option 3: Pb-208**
- Protons (Z) = 82 (which is a magic number)
- Neutrons (N) = 208 - 82 = 126 (which is a magic number)
- Conclusion: Satisfies both conditions.

- **Option 4: Sn-118**
- Protons (Z) = 50 (which is a magic number)
- Neutrons (N) = 118 - 50 = 68 (not a magic number)
- Conclusion: Does not satisfy both conditions.

3. **Final Conclusion**:
- The only nuclide that has magic numbers for both protons and neutrons is **Pb-208**.

### Answer:
**Pb-208** (Option 3) has magic numbers for both protons and neutrons.

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