1. The problem asks for the possible values of the remainder when a positive integer is divided by 4. 2. When dividing by 4, the possible remainders are the integers from 0 up to one less than 4. 3. This is because dividing by 4 means we split the number into groups of 4, so the leftover part (remainder) cannot be equal to or greater than 4. 4. Therefore, the possible remainders are $0, 1, 2,$ or $3$. Final Answer: The possible values of the remainder when a positive integer is divided by 4 are $0, 1, 2,$ or $3$.