Which of the following numbers are solutions to the system of simultaneous linear congruences: n≡2(mod3)
, n≡1(mod16)
?

17

49

97

113

209

257

To solve the system of simultaneous linear congruences:

1. \( n \equiv 2 \pmod{3} \)
2. \( n \equiv 1 \pmod{16} \)

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

### Checking the numbers:

1. **17:**
   - \( 17 \mod 3 = 2 \) (Satisfies \( n \equiv 2 \pmod{3} \))
   - \( 17 \mod 16 = 1 \) (Satisfies \( n \equiv 1 \pmod{16} \))
   - **17** is a solution.

2. **49:**
   - \( 49 \mod 3 = 1 \) (Does not satisfy \( n \equiv 2 \pmod{3} \))
   - \( 49 \mod 16 = 1 \) (Satisfies \( n \equiv 1 \pmod{16} \))
   - **49** is **not** a solution.

3. **97:**
   - \( 97 \mod 3 = 1 \) (Does not satisfy \( n \equiv 2 \pmod{3} \))
   - \( 97 \mod 16 = 1 \) (Satisfies \( n \equiv 1 \pmod{16} \))
   - **97** is **not** a solution.

4. **113:**
   - \( 113 \mod 3 = 2 \) (Satisfies \( n \equiv 2 \pmod{3} \))
   - \( 113 \mod 16 = 1 \) (Satisfies \( n \equiv 1 \pmod{16} \))
   - **113** is a solution.

5. **209:**
   - \( 209 \mod 3 = 2 \) (Satisfies \( n \equiv 2 \pmod{3} \))
   - \( 209 \mod 16 = 1 \) (Satisfies \( n \equiv 1 \pmod{16} \))
   - **209** is a solution.

6. **257:**
   - \( 257 \mod 3 = 2 \) (Satisfies \( n \equiv 2 \pmod{3} \))
   - \( 257 \mod 16 = 1 \) (Satisfies \( n \equiv 1 \pmod{16} \))
   - **257** is a solution.

### Solutions:
The numbers that satisfy both congruences are:
- **17**
- **113**
- **209**
- **257**

Would you like to go into more details, or do you have any questions?

### Related Questions:
1. What is the Chinese Remainder Theorem and how does it apply here?
2. How can you verify a solution to a system of congruences?
3. What is the significance of moduli in congruences?
4. How would you solve these congruences if the moduli were not coprime?
5. Can you explain the difference between modular arithmetic and regular arithmetic?

### Tip:
When dealing with simultaneous congruences, always check each condition separately and ensure both are satisfied by the solution.

Explore solutions to simultaneous linear congruences where n ≡ 2 (mod 3) and n ≡ 1 (mod 16). Check which numbers among 17, 49, 97, 113, 209, 257 satisfy both congruences with detailed step-by-step verification. Understand the application of modular arithmetic and the Chinese Remainder Theorem.

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