Factors and Multiples - Factors and Multiples

Factors are the numbers that are multiplied together to get a product. For example, in $3 \times 4 = 12$, both $3$ and $4$ are factors of $12$. Visually, if you have $12$ beads, factors represent the different ways you can arrange them into complete rectangular rows and columns, such as $2$ rows of $6$ or $3$ rows of $4$.

Properties of Factors: $1$ is a factor of every number, and every number is a factor of itself. A factor of a number is always less than or equal to that number. For instance, the factors of $6$ are $1, 2, 3,$ and $6$, all of which are smaller than or equal to $6$.

Multiples are the products obtained by multiplying a number by other counting numbers like $1, 2, 3, \dots$. For example, the multiples of $5$ are $5, 10, 15, 20,$ and so on. Visually, multiples can be thought of as equal jumps on a number line; starting at $0$, a jump of $4$ units lands you on $4, 8, 12,$ and $16$, which are all multiples of $4$.

Properties of Multiples: Every number is a multiple of $1$ and itself. Unlike factors, the number of multiples for any given number is infinite. A multiple of a number is always greater than or equal to the number itself.

Even and Odd Numbers: Numbers that are multiples of $2$ are called even numbers (ending in $0, 2, 4, 6,$ or $8$). Numbers that are not multiples of $2$ are called odd numbers (ending in $1, 3, 5, 7,$ or $9$). Visually, even numbers can always be grouped into pairs with nothing left over, while odd numbers will always have one single unit remaining after pairing.

Common Factors are factors that are shared by two or more numbers. For example, factors of $8$ are $\{1, 2, 4, 8\}$ and factors of $12$ are $\{1, 2, 3, 4, 6, 12\}$. The common factors are $1, 2,$ and $4$.

Common Multiples are numbers that appear in the lists of multiples for two or more different numbers. For example, multiples of $3$ are $3, 6, 9, 12, \dots$ and multiples of $4$ are $4, 8, 12, 16, \dots$. A common multiple of $3$ and $4$ is $12$.