Computation and Operations - Factors and Multiples

Factors are whole numbers that multiply together to get a specific product. For example, in the equation $3 \times 4 = 12$, both $3$ and $4$ are factors of $12$. Visually, factors represent the possible side lengths of a rectangle made from a specific number of square tiles.

A Factor Rainbow is a visual way to list all factor pairs of a number. You list the numbers in order from smallest to largest and draw an arc connecting the pairs that multiply to the target number. For $12$, arcs would connect $1$ and $12$, $2$ and $6$, and $3$ and $4$.

Multiples are the products of a number and any whole number (like $1, 2, 3, ...$). You can think of multiples as 'skip-counting' results. On a number line, multiples of $5$ appear as equal-sized jumps landing on $5, 10, 15, 20$, and so on.

Prime Numbers are numbers greater than $1$ that have exactly two factors: $1$ and the number itself. Visually, a prime number of dots (like $7$) can only be arranged in a single straight line ($1 \times 7$), never in a wider rectangle.

Composite Numbers are numbers that have more than two factors. Unlike prime numbers, composite numbers can be arranged into multiple different rectangular arrays. For example, $12$ dots can be arranged as $1 \times 12$, $2 \times 6$, or $3 \times 4$.

Common Factors are factors that are shared by two or more numbers. If you draw a Venn diagram with two circles representing the factors of $8$ and the factors of $12$, the numbers $1, 2,$ and $4$ would be placed in the overlapping center section.

Common Multiples are numbers that appear in the skip-counting sequences of two or more different numbers. For example, if you look at a hundred-chart and circle multiples of $3$ and shade multiples of $4$, the squares that are both circled and shaded (like $12$ and $24$) are the common multiples.

Divisibility Rules help identify factors quickly. For example, a number is divisible by $2$ if it is even (ends in $0, 2, 4, 6, 8$) and divisible by $5$ if it ends in $0$ or $5$. On a $10$-column number grid, multiples of $10$ form a straight vertical line in the last column.