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.