Final answer:
Factor pairs of a number start to repeat when the number is a perfect square. That's because perfect squares have a square root that is an integer, which is one of the factors that appears twice, leading to repetition in factor pairs.
Step-by-step explanation:
The question "When do the factor pairs of a number start to repeat?" refers to the concept of identifying the pairs of numbers that can be multiplied together to give the original number. For example, the factor pairs of 8 are (1, 8) and (2, 4). Notice that there is no need to include (4, 2) as a separate pair since (2, 4) already represents that combination.
Factor pairs start to repeat when you reach a factor that is repeated, which happens in the case of perfect squares. A perfect square is a number that is the square of an integer. For example, 9 is a perfect square because it is 3 times 3. Its factor pairs are (1,9) and (3,3). After (3,3), we would be repeating the earlier factor in reverse order, (9,1), which we don't count again.
The correct answer to the question is C) When the number is a perfect square. Options A, B, and D are not when factor pairs repeat. This aligns with the multiplication principle where M = b^n (M is the product, b is the base, and n is the power), but instead of using exponents, we are simply looking for pairs of integers (factor pairs) where their product is the original number. Notice that when a number is a perfect square, its square root is one of the factors that appears twice, hence leading to repeated factor pairs if you were to continue listing them beyond that point.