Final answer:
When a whole number is squared and divided by 5, the remainder is always 0, 1, or 4 due to the properties of whole numbers and their remainders when squared.
Step-by-step explanation:
Proving Remainders of Squares Divided by 5
When you square a whole number and divide it by 5, the possible remainders are always 0, 1, or 4. This is a consequence of the fact that any whole number can be written as either 5k, 5k+1, 5k+2, 5k+3, or 5k+4 for some integer k. When we square these, we get 25k², (5k+1)², (5k+2)², (5k+3)², and (5k+4)², respectively. Simplifying these expressions, we see that the term involving k is always divisible by 5, and the constant term is the square of the remainder when the original number is divided by 5. This results in remainders of 0, 1, 4, 9, and 16 when divided by 5, which further simplifies to remainders of 0, 1, and 4 respectively because 9 and 16 also give remainders of 4 when divided by 5. Thus, no matter what whole number you start with, squaring it and dividing by 5 will always result in a remainder of 0, 1, or 4.