Factors, Multiples, and Primes

Prerequisites

Arithmetic | 8 of 17

Prerequisites: Multiplication, Division. Enables: Fractions.

Factors

A factor divides a number evenly.

$$ a = bc $$

then $b$ and $c$ are factors of $a$.

Example:

24 = 6 \cdot 4

so $6$ and $4$ are factors of $24$.

A multiple of a number is the product of that number and an integer.

Multiples of $6$ include:

6,\ 12,\ 18,\ 24,\ 30,\ \ldots

A prime number has exactly two positive factors: $1$ and itself.

Examples:

2,\ 3,\ 5,\ 7,\ 11,\ 13

A composite number has more than two positive factors.

Examples:

4,\ 6,\ 8,\ 9,\ 10,\ 12

The number $1$ is neither prime nor composite.

Prime factorization writes a number as a product of primes.

Example:

84 = 2 \cdot 42

84 = 2 \cdot 2 \cdot 21

84 = 2^2 \cdot 3 \cdot 7

The greatest common factor, or GCF, is the largest factor shared by two or more numbers.

Example:

Factors of $18$:

1,\ 2,\ 3,\ 6,\ 9,\ 18

Factors of $24$:

1,\ 2,\ 3,\ 4,\ 6,\ 8,\ 12,\ 24

Therefore:

$$ GCF(18,24)=6 $$

The least common multiple, or LCM, is the smallest positive multiple shared by two or more numbers.

Example:

Multiples of $6$:

6,\ 12,\ 18,\ 24,\ 30,\ldots

Multiples of $8$:

8,\ 16,\ 24,\ 32,\ldots

Therefore:

$$ LCM(6,8)=24 $$