The correct answer is **A) $16$**. We need to find the remainder when $13^{2024}$ is divided by $99$. This can be written as finding $13^{2024} \pmod{99}$. **Step 1: Factorize the modulus and apply Euler's Totient Theorem.** The modulus is $N=99$. We can factorize $99$ as $99 = 9 \times 11$. Since $\gcd(13, 99) = 1$, we can use Euler's Totient Theorem, which states that if $\gcd(a, n) = 1$, then $a^{\phi(n)} \equiv 1 \pmod n$. First, calculate Euler's Totient function for $99$: $\phi(99) = \phi(9 \times 11) = \phi(3^2 \times 11)$. Since $9$ and $11$ are coprime, $\phi(99) = \phi(9) \times \phi(11)$. $\phi(3^2) = 3^2 - 3^1 = 9 - 3 = 6$. $\phi(11) = 11 - 1 = 10$. So, $\phi(99) = 6 \times 10 = 60$. According to Euler's Totient Theorem, $13^{60} \equiv 1 \pmod{99}$. **Step 2: Reduce the exponent modulo $\phi(99)$.** We need to find the remainder of the exponent $2024$ when divided by $60$. $2024 \div 60$: $2024 = 60 \times 33 + 44$. So, $2024 \equiv 44 \pmod{60}$. Therefore, $13^{2024} \equiv 13^{60 \times 33 + 44} \equiv (13^{60})^{33} \times 13^{44} \equiv 1^{33} \times 13^{44} \equiv 13^{44} \pmod{99}$. **Step 3: Calculate $13^{44} \pmod{99}$ using the Chinese Remainder Theorem (CRT).** We need to find $x$ such that $x \equiv 13^{44} \pmod{99}$. This is equivalent to solving the system of congruences: 1. $x \equiv 13^{44} \pmod 9$ 2. $x \equiv 13^{44} \pmod {11}$ **For Congruence 1 (modulo 9):** $13 \equiv 4 \pmod 9$. So, $13^{44} \equiv 4^{44} \pmod 9$. We calculate $\phi(9) = 6$. So $4^6 \equiv 1 \pmod 9$ (since $\gcd(4,9)=1$). Reduce the exponent $44$ modulo $6$: $44 = 6 \times 7 + 2$. So $44 \equiv 2 \pmod 6$. Therefore, $4^{44} \equiv 4^2 \equiv 16 \pmod 9$. $16 \equiv 7 \pmod 9$. So, $x \equiv 7 \pmod 9$. **For Congruence 2 (modulo 11):** $13 \equiv 2 \pmod {11}$. So, $13^{44} \equiv 2^{44} \pmod {11}$. We calculate $\phi(11) = 10$. So $2^{10} \equiv 1 \pmod {11}$. Reduce the exponent $44$ modulo $10$: $44 = 10 \times 4 + 4$. So $44 \equiv 4 \pmod {10}$. Therefore, $2^{44} \equiv 2^4 \equiv 16 \pmod {11}$. $16 \equiv 5 \pmod {11}$. So, $x \equiv 5 \pmod {11}$. **Step 4: Solve the system of congruences using CRT.** We have: 1. $x \equiv 7 \pmod 9$ 2. $x \equiv 5 \pmod {11}$ From congruence (1), $x = 9k + 7$ for some integer $k$. Substitute this into congruence (2): $9k + 7 \equiv 5 \pmod {11}$ $9k \equiv 5 - 7 \pmod {11}$ $9k \equiv -2 \pmod {11}$ To make the right side positive, add $11$: $9k \equiv -2 + 11 \pmod {11}$ $9k \equiv 9 \pmod {11}$ Since $\gcd(9, 11) = 1$, we can divide by $9$ on both sides: $k \equiv 1 \pmod {11}$ The smallest non-negative value for $k$ is $1$. Substitute $k=1$ back into $x = 9k + 7$: $x = 9(1) + 7 = 9 + 7 = 16$. Thus, the remainder when $13^{2024}$ is divided by $99$ is $16$. The final answer is $\boxed{\text{16}}$.