Which of the following numbers are solutions to the system of simultaneous linear congruences: n≡2(mod5)
, n≡3(mod7)
?

87
 

122
 

147
 

191
 

227
 

257

To solve the system of simultaneous linear congruences:

1. \( n \equiv 2 \pmod{5} \)
2. \( n \equiv 3 \pmod{7} \)

We need to check which of the given numbers satisfy both congruences.

### Checking the numbers:

1. **87:**
   - \( 87 \mod 5 = 2 \) (Satisfies \( n \equiv 2 \pmod{5} \))
   - \( 87 \mod 7 = 3 \) (Satisfies \( n \equiv 3 \pmod{7} \))
   - **87** is a solution.

2. **122:**
   - \( 122 \mod 5 = 2 \) (Satisfies \( n \equiv 2 \pmod{5} \))
   - \( 122 \mod 7 = 3 \) (Satisfies \( n \equiv 3 \pmod{7} \))
   - **122** is a solution.

3. **147:**
   - \( 147 \mod 5 = 2 \) (Satisfies \( n \equiv 2 \pmod{5} \))
   - \( 147 \mod 7 = 0 \) (Does not satisfy \( n \equiv 3 \pmod{7} \))
   - **147** is **not** a solution.

4. **191:**
   - \( 191 \mod 5 = 1 \) (Does not satisfy \( n \equiv 2 \pmod{5} \))
   - \( 191 \mod 7 = 2 \) (Does not satisfy \( n \equiv 3 \pmod{7} \))
   - **191** is **not** a solution.

5. **227:**
   - \( 227 \mod 5 = 2 \) (Satisfies \( n \equiv 2 \pmod{5} \))
   - \( 227 \mod 7 = 3 \) (Satisfies \( n \equiv 3 \pmod{7} \))
   - **227** is a solution.

6. **257:**
   - \( 257 \mod 5 = 2 \) (Satisfies \( n \equiv 2 \pmod{5} \))
   - \( 257 \mod 7 = 5 \) (Does not satisfy \( n \equiv 3 \pmod{7} \))
   - **257** is **not** a solution.

### Solutions:
The numbers that satisfy both congruences are:
- **87**
- **122**
- **227**

Would you like more details or have any other questions?

### Related Questions:
1. How can you solve systems of congruences with more than two equations?
2. What methods exist to solve simultaneous congruences if the moduli are not coprime?
3. What is the general solution for \( n \) if the moduli are coprime?
4. How does the Chinese Remainder Theorem simplify solving such systems?
5. What are the applications of modular arithmetic in real-world problems?

### Tip:
When solving simultaneous congruences, try to reduce the problem to simpler cases by testing the conditions on smaller numbers before checking larger ones.

Explore how to solve the system of simultaneous linear congruences n≡2(mod5) and n≡3(mod7) using modular arithmetic and the Chinese Remainder Theorem. Check which numbers satisfy both congruences among 87, 122, 147, 191, 227, and 257.

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