Finding All the Factors of a Number

There are often several ways to talk about the same idea. So far, we’ve seen that if \(m\) is a multiple of \(n,\) we can say that \(m\) is divisible by \(n.\) We know that \(72\) is the product of \(8\) and \(9,\) so we can say \(72\) is a multiple of \(8\) and \(72\) is a multiple of \(9.\) We can also say \(72\) is divisible by \(8\) and by \(9.\) Another way to talk about this is to say that \(8\) and \(9\) are factors of \(72.\) When we write \(72=8\cdot 9\) we can say that we have factored \(72.\)

The image shows the equation 8 times 9 equals 72. The 8 and 9 are labeled as factors and the 72 is labeled product.

Factors

If \(a\cdot b=m,\) then \(a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b\) are factors of \(m,\) and \(m\) is the product of \(a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b.\)

In algebra, it can be useful to determine all of the factors of a number. This is called factoring a number, and it can help us solve many kinds of problems.For example, suppose a choreographer is planning a dance for a ballet recital. There are \(24\) dancers, and for a certain scene, the choreographer wants to arrange the dancers in groups of equal sizes on stage.

In how many ways can the dancers be put into groups of equal size? Answering this question is the same as identifying the factors of \(24.\) The table summarizes the different ways that the choreographer can arrange the dancers.

Number of Groups	Dancers per Group	Total Dancers
\(1\)	\(24\)	\(1\cdot 24=24\)
\(2\)	\(12\)	\(2\cdot 12=24\)
\(3\)	\(8\)	\(3\cdot 8=24\)
\(4\)	\(6\)	\(4\cdot 6=24\)
\(6\)	\(4\)	\(6\cdot 4=24\)
\(8\)	\(3\)	\(8\cdot 3=24\)
\(12\)	\(2\)	\(12\cdot 2=24\)
\(24\)	\(1\)	\(24\cdot 1=24\)

What patterns do you see in the table above? Did you notice that the number of groups times the number of dancers per group is always \(24?\) This makes sense, since there are always \(24\) dancers.

You may notice another pattern if you look carefully at the first two columns. These two columns contain the exact same set of numbers—but in reverse order. They are mirrors of one another, and in fact, both columns list all of the factors of \(24,\) which are:

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

We can find all the factors of any counting number by systematically dividing the number by each counting number, starting with \(1.\) If the quotient is also a counting number, then the divisor and the quotient are factors of the number. We can stop when the quotient becomes smaller than the divisor.

Find all the factors of a counting number.

Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor.
1. If the quotient is a counting number, the divisor and quotient are a pair of factors.
2. If the quotient is not a counting number, the divisor is not a factor.
List all the factor pairs.
Write all the in order from smallest to largest.

Example

Find all the factors of \(72.\)

Solution

Divide \(72\) by each of the counting numbers starting with \(1.\) If the quotient is a whole number, the divisor and quotient are a pair of factors.

The figure shows a table with ten rows and four columns. The first row is a header row and labels the rows “Dividend”, “Divisor”, “Quotient”, and “Factors”. Under the “Dividend” column all rows show the number 72. In the second row the “Divisor” column is 1, the “Quotient” column is 72 and the “Factors” column is 1 and 72. In the third row the “Divisor” column is 2, the “Quotient” column is 36 and the “Factors” column is 2 and 36. In the fourth row the “Divisor” column is 3, the “Quotient” column is 24 and the “Factors” column is 3 and 24. In the fifth row the “Divisor” column is 4, the “Quotient” column is 18 and the “Factors” column is 4 and 18. In the sixth row the “Divisor” column is 5, the “Quotient” column is 14.4 and the “Factors” column is blank. In the seventh row the “Divisor” column is 6, the “Quotient” column is 12 and the “Factors” column is 6 and 12. In the eighth row the “Divisor” column is 7, the “Quotient” column is about 10.29 and the “Factors” column is blank. In the ninth row the “Divisor” column is 8, the “Quotient” column is 9 and the “Factors” column is 8 and 9. In the tenth row the “Divisor” column is 9, the “Quotient” column is 8 and the “Factors” column is 9 and 8.

The next line would have a divisor of \(9\) and a quotient of \(8.\) The quotient would be smaller than the divisor, so we stop. If we continued, we would end up only listing the same factors again in reverse order. Listing all the factors from smallest to greatest, we have

\(1,2,3,4,6,8,9,12,18,24,36,\text{and}\phantom{\rule{0.2em}{0ex}}72\)

Optional Videos

Factors

Example 1: Determine Factors of a Number

Example 2: Determine Factors of a Number

Example 3: Determine Factors of a Number

Finding All the Factors of a Number