We are asked to find the integer $c$ such that $0 \leq c \leq 12$ and $c \equiv a + b \pmod{13}$. Given that:

• $a \equiv 4 \pmod{13}$
• $b \equiv 9 \pmod{13}$

Step 1: Add $a$ and $b$

$a + b = 4 + 9 = 13$

Step 2: Find $c$ modulo 13

Since we are asked to find $c \equiv a + b \pmod{13}$, we compute $13 \mod 13$: $13 \div 13 = 1 \text{ with a remainder of } 0$ Thus, $13 \equiv 0 \pmod{13}$.

Conclusion

The integer $c$ that satisfies $c \equiv a + b \pmod{13}$ and $0 \leq c \leq 12$ is $c = 0$.

Would you like further details or have any questions?

Here are 5 related questions:

1. What is the general method to add numbers modulo $n$?
2. How would you solve $c \equiv a - b \pmod{13}$?
3. How does modular addition differ from modular multiplication?
4. Can you explain how to handle negative numbers in modular arithmetic?
5. How do you solve systems of congruences like the Chinese Remainder Theorem?

Tip: In modular arithmetic, adding numbers first and then reducing modulo $n$ is equivalent to reducing each number modulo $n$ first, then adding the results.