Arrays and Factors

An array is a way of arranging objects, squares, or dots into equal rows and equal columns, forming a neat rectangle. Arrays are a visual tool that connects multiplication and division, because the total number of items in an array always equals the number of rows multiplied by the number of columns.

For example, 12 counters can be arranged in an array with 3 rows and 4 columns, since \( 3 \times 4 = 12 \).

Since an array's total always equals rows times columns, every array you can build for a number shows one of its factor pairs. If a number can be arranged into a rectangle with whole-number rows and columns, then both of those numbers are factors of it.

The same 12 counters used above can also form an array with 2 rows and 6 columns, since \( 2 \times 6 = 12 \) as well.

Notice that swapping the rows and columns (\( 3 \times 4 \) versus \( 4 \times 3 \)) gives the same factor pair, just rotated. This is a helpful visual reminder that multiplication order does not change the product.

To find every factor of a number using arrays, try building rectangles with different numbers of rows, starting at 1 and working up. Stop once the row number and column number meet in the middle, since after that point the pairs start repeating in reverse order.

For 12, the complete list of arrays gives these factor pairs:

• \( 1 \times 12 \)
• \( 2 \times 6 \)
• \( 3 \times 4 \)

So the factors of 12 are 1, 2, 3, 4, 6, and 12. This same idea of breaking a number into rows and columns is closely related to the area model you use when dividing using area models, since both methods rely on rectangles built from equal groups.

Some numbers can only be arranged into one array shape (other than a single row or column). Try building an array for 7. You will find that the only possible rectangle is 1 row of 7, since \( 1 \times 7 = 7 \) and no other whole-number rows and columns work.

A number like 7, which can only form one array besides a single line, is called a prime number. A number like 12, which can form several different arrays, is called a composite number. Counting the number of possible arrays is a quick, visual way to decide whether a number is prime or composite.

Find all the factors of 18 using arrays.

1. Start with 1 row: \( 1 \times 18 = 18 \), so 1 and 18 are factors.
2. Try 2 rows: \( 2 \times 9 = 18 \), so 2 and 9 are factors.
3. Try 3 rows: \( 3 \times 6 = 18 \), so 3 and 6 are factors.
4. Try 4 rows: 18 cannot be split evenly into 4 rows, so 4 is not a factor.
5. Try 5 rows: 18 cannot be split evenly into 5 rows either, so 5 is not a factor.
6. Once you reach 6 rows, the pairs begin repeating (\( 6 \times 3 \)), so you have found them all.

The factors of 18 are 1, 2, 3, 6, 9, and 18.

Finding factors with arrays builds a strong foundation for many later skills, including simplifying fractions, finding common factors, and understanding multiplication and division facts together. Once you are comfortable with smaller arrays, the same rectangle thinking extends naturally to larger calculations, such as multiplying multi-digit numbers and dividing multi-digit numbers, where numbers are broken into friendly parts, just like the rows and columns of an array.

Related lessons

• Math 5