Question

Rob says all numbers have an even number of factors. Kayla says some numbers have an odd number of factors who is correct explain

Answer 1

Rob is correct. All numbers have an even number of factors except for perfect squares, which have an odd number of factors.

Let's break down the explanation:

1. Factors of a Number:

The factors of a number are the numbers that can evenly divide that number. For example, the factors of 6 are 1, 2, 3, and 6.

2. Counting Factors:

When we count the factors of a number, we can pair them up. For each factor
`\(a\)` of the number, there is a corresponding factor
`\(b\)` such that
`\(a * b\)` equals the number. For example, for 6, the pairs are (1, 6) and (2, 3).

3. Even Number of Factors:

In general, unless the number is a perfect square, every factor has a corresponding factor, resulting in an even number of factors. For example:

- For 6: (1, 6) and (2, 3), giving a total of 4 factors (even).

- For 12: (1, 12), (2, 6), and (3, 4), giving a total of 6 factors (even).

4. Odd Number of Factors (Perfect Squares):

When a number is a perfect square, one of its factors
`(\(a\))` is equal to its square root
`(\(√(a)\))`. In this case, there is no corresponding factor, and the factor is not paired. For example:

- For 9: (1, 9), giving a total of 3 factors (odd).

So, while Rob is generally correct, Kayla is correct in pointing out the exception that perfect squares have an odd number of factors.