Suppose that a and b are integers, a ≡ 4 (mod 13), and
b ≡ 9 (mod 13). Find the integer c with 0 ≤ c ≤ 12 such
that: c ≡ a + b (mod 13)

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.

Suppose that a and b are integers, a ≡ 4 (mod 13), and b ≡ 9 (mod 13). Find the integer c with 0 ≤ c ≤ 12 such that: c ≡ a + b (mod 13).

Learn how to solve a modular arithmetic problem where a ≡ 4 (mod 13) and b ≡ 9 (mod 13). This problem covers key concepts of modular addition and congruence relations, with step-by-step solutions suitable for students in Grades 9-12.

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