Lesson 2: Factors

Property

A multiplication equation can be represented by an array.
If the first factor is the number of groups and the second factor is the size of each group, the array will have a number of rows equal to the first factor and a number of columns equal to the second factor.

Examples

Property

The order of factors does not change the product. This is known as the commutative property of multiplication.

Examples

• For the number 16, the factor pairs (1, 16) and (16, 1) both result in the product 16: $1 \times 16 = 16$ and $16 \times 1 = 16$.
• For the number 12, the factor pairs (3, 4) and (4, 3) both result in the product 12: $3 \times 4 = 12$ and $4 \times 3 = 12$.

Explanation

The commutative property of multiplication states that you can multiply numbers in any order and the result will be the same. This means that if you find one factor pair for a number, like (2, 9) for 18, you can switch the order to find another, (9, 2). This is useful because once you find one factor pair, you automatically know a second one just by swapping the numbers. This can make finding all the factors of a number faster.

Property

If $a \cdot b = m$, and both $a$ and $b$ are integers, then $a$ and $b$ are factors of $m$, and $m$ is the product of $a$ and $b$.

To find all the factors of a counting number:

1. Divide the number by each counting number (1, 2, 3, ...), in order, until the quotient is smaller than the divisor. If the quotient is a counting number, the divisor and quotient are a factor pair.
2. List all the factor pairs.
3. Write all the factors in order from smallest to largest.

Examples

• To find all the factors of 24, we look for pairs of numbers that multiply to 24: $1 \cdot 24$, $2 \cdot 12$, $3 \cdot 8$, and $4 \cdot 6$. So, the factors are 1, 2, 3, 4, 6, 8, 12, and 24.