Which of the following can be solved using CRT? Why?
a. � ≡ 2(��� 4), � ≡ 3(��� 6) c. � ≡ 1(��� 8), � ≡ 2(��� 16)
b. � ≡ 5(��� 7), � ≡ 3(��� 11) d. � ≡ 0(��� 9), � ≡ 3(��� 12)

The **Chinese Remainder Theorem (CRT)** can be applied to systems of congruences when the moduli (the numbers in parentheses) are **pairwise coprime**, meaning each pair of moduli has a greatest common divisor (GCD) of 1.

Let's analyze each system of congruences:

### a. \( x \equiv 2 \pmod{4}, \ x \equiv 3 \pmod{6} \)

- **Moduli**: 4 and 6
- **GCD(4, 6) = 2**, which is not 1. Since the moduli are not coprime, **CRT cannot be applied**.

### b. \( x \equiv 5 \pmod{7}, \ x \equiv 3 \pmod{11} \)

- **Moduli**: 7 and 11
- **GCD(7, 11) = 1**, meaning the moduli are coprime.
- Since the moduli are coprime, **CRT can be applied** to solve this system.

### c. \( x \equiv 1 \pmod{8}, \ x \equiv 2 \pmod{16} \)

- **Moduli**: 8 and 16
- **GCD(8, 16) = 8**, which is not 1. Since the moduli are not coprime, **CRT cannot be applied**.

### d. \( x \equiv 0 \pmod{9}, \ x \equiv 3 \pmod{12} \)

- **Moduli**: 9 and 12
- **GCD(9, 12) = 3**, which is not 1. Since the moduli are not coprime, **CRT cannot be applied**.

### Conclusion:
Only **system b** can be solved using the Chinese Remainder Theorem because the moduli 7 and 11 are coprime.

Would you like further details or have any questions? 

Here are 5 related questions:
1. How do you solve a system of congruences using CRT?
2. What happens if the moduli are not coprime?
3. Can CRT be applied if one modulus is a multiple of the other?
4. What are some real-life applications of CRT?
5. How can you check if two numbers are coprime?

**Tip**: To check if numbers are coprime, find their GCD. If the GCD is 1, they are coprime.

Which of the following can be solved using CRT? Why?
a. x ≡ 2 (mod 4), x ≡ 3 (mod 6)
b. x ≡ 5 (mod 7), x ≡ 3 (mod 11)
c. x ≡ 1 (mod 8), x ≡ 2 (mod 16)
d. x ≡ 0 (mod 9), x ≡ 3 (mod 12)

Explore the application of the Chinese Remainder Theorem to different systems of congruences. Learn which systems of modular arithmetic can be solved using CRT and why. This content covers modular arithmetic, GCD, and the conditions necessary for applying CRT.

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