Lesson 6: Divisors

Property

A number is a multiple of $n$ if it is the product of a counting number and $n$.
If a number $m$ is a multiple of $n$, then $m$ is divisible by $n$.
If $a \cdot b = m$, and both $a$ and $b$ are integers, then $a$ and $b$ are factors of $m$. We also say that $a$ and $b$ are divisors of $m$.
A composite number is a counting number greater than 1 that has more than two factors.

Examples

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.

Property

If $a$ divides $b$ and $b$ divides $c$, then $a$ divides $c$. In mathematical notation: if $a \mid b$ and $b \mid c$, then $a \mid c$.

Examples