Problem 1 - Entrance Test

We use Fermat's Little Theorem, which states that if p is a prime number, then for any integer a not divisible by p, a^(p-1) is congruent to 1 (mod p). Here, p = 23 and a = 3. Since 23 is a prime number and 3 is not divisible by 23, we have 3^(23-1) = 3^22 is congruent to 1 (mod 23). Now we need to find the remainder of the exponent 2023 when divided by 22: 2023 = 22 * 92 + 9. So, 3^2023 = 3^(22*92 + 9) = (3^22)^92 * 3^9 (mod 23). Since 3^22 is congruent to 1 (mod 23), we have: (3^22)^92 * 3^9 is congruent to 1^92 * 3^9 (mod 23). This simplifies to 3^9 (mod 23). Now we calculate powers of 3 modulo 23: 3^1 = 3 3^2 = 9 3^3 = 27 which is congruent to 4 (mod 23) 3^4 = 3^3 * 3 is congruent to 4 * 3 = 12 (mod 23) 3^5 = 12 * 3 = 36 which is congruent to 13 (mod 23) 3^6 = 13 * 3 = 39 which is congruent to 16 (mod 23) 3^7 = 16 * 3 = 48 which is congruent to 2 (mod 23) 3^8 = 2 * 3 = 6 (mod 23) 3^9 = 6 * 3 = 18 (mod 23). Thus, the remainder when 3^2023 is divided by 23 is 18.