Final answer:
The congruence class [4]m is [0]m when m=2 and [1]m when m=3. It represents the set of all integers that are congruent to 4 modulo m. The correct answer is c) [2]m, [1]m.
Step-by-step explanation:
The student is asking about congruence classes, which are sets of numbers where each number is congruent to a particular integer modulo m, where m is a fixed positive integer known as the modulus. In this context, the congruence class [4]m represents all integers that leave the same remainder when divided by m as 4 does. To find the congruence class [4]m for different values of m:
a) When m is 2, we consider the remainder when 4 is divided by 2. Since 4 is divisible by 2, the remainder is 0. Thus, the congruence class [4] when m is 2 is [0]m, which means answer choice a) [0]m is correct for part a).
b) When m is 3, we consider the remainder when 4 is divided by 3. Since 4 divided by 3 leaves a remainder of 1, the congruence class [4] when m is 3 is [1]m, which means answer choice c) [1]m is correct for part b).