Identify Multiples and Factors, Including Finding All Factor Pairs of a Number and Common Factors of Two Numbers - Reasoning

The concept of factors and multiples is a fundamental aspect of multiplication and division in mathematics. In a reasoning exercise, the statement is made that all numbers have an even number of factors because factors come in pairs. However, this is not always the case. Tia's assertion is challenged with the example of certain numbers, such as 9 and 25, which have an odd number of factors. The reason for this is due to the fact that these numbers are perfect squares, and one of their factor pairs consists of the same number appearing twice. For instance, the number 9 can be factored as 1 x 9 and 3 x 3. Here, the pair 3 x 3 is counted as one factor because it is the same number multiplied by itself, resulting in an odd number of factors.

The exercise encourages students to explore and prove whether Tia's statement is true or not. To prove it, one must consider the factors of various numbers and identify their pairs. While most numbers do have factors that come in distinct pairs, leading to an even count, perfect squares are the exception to this rule. When students engage in this reasoning task, they are expected to use their understanding of multiplication, division, and the properties of numbers to investigate and establish the validity of the claim. Through this process, they learn that while many numbers do have an even number of factors, not all do, due to the unique factor pairs of perfect squares.