Factors of 72—with division and prime factorization

Factors of 72

Factors Of 72

The factors of 72 are the numbers that result in 72 when multiplied together in pairs

For example

2 x 3 = 6, which states that 2 and 3 are factors of 6. A multiple of 72 is usually an extended version of 72, such as 72, 144, 216, 288, 360, 432, 504, 576, 648, etc. We will use the factorization method to find the factors of 72. 

It is possible to represent the factor 72 as a positive or negative number.

However,  factors 72 cannot be a decimal or fraction. For example, for factors of 72, the factors would be (1, 72) or (-1, 72). When we multiply a pair of negative numbers, such as multiplying -1 by -72, it results in 72.

Factors of 72

Using the prime factorization method, we will learn the factors of 72, the pair factors, and the prime factors of 72.

What are the Factors of 72?

In mathematics, factors of 72 are the numbers that divide it exactly without leaving a remainder.

Therefore, the 72 factors are the numbers multiplied in pairs that result in a number of 72. As the number 72 has several factors other than one and the number itself, being a composite number.

Therefore, the factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. 

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.

Prime Factorization of 72: 2 × 2 × 2 × 3 × 3 or 23 × 32

What are the Factors of -72?

When we multiply two numbers, and we get -4 as a product, that numbers are the factors of -4. 

Here is the explanation:


-1 × 72 = -72

 1 × -72 = -72

-2 × 36 = -72  

2 × -36 = -72 

-3 × 24 = -72 

3 × -24 = -72 

-4 × 18 = -72 

4 × -18 = -72

-6 × 12 = -72 

6 × -12 = -72 

-8 × 9 = -72  

8 × -9 = -72


We can get -72 by having (-1, 72), (-2, 36), (-3, 24), (-4, 18), (-6, 12), and(-8, 9)as a pair of factors. Similarly, we can also get -72 by having (1, -72), (2, -36), (3, -24), (4, -18), (6, 12), and (8, -9) as a pair of factors.

Pair Factors of 72

Pair factors of 72 are the two numbers multiplied together to give 72 as the result. Positive or negative pair factors of 72 are possible. The result of multiplying the pair of negative numbers is 72. 

Here are the negative and positive factors of 72:

Positive Pair Factors of 72:

Positive Factors of 72Positive Pair Factors of 72
1 × 72 (1, 72)
2 × 36 (2, 36)
3 × 24 (3, 24)
4 × 18 (4, 18)
6 × 12(6, 12)
8 × 9(8, 9)

As a result, the positive pair factors are

(1, 72), (2, 36), (3, 24), (4, 18), (6, 12) and (8, 9).

Negative Pair Factors of 72:

Negative Factors of 72Negative Pair Factors of 72
-1 × -72 (-1, -72)
-2 × -36 (-2, -36)
-3 × -24 (-3, -24)
-4 × -18 (-4, -18)
-6 × -12(-6, -12)
-8 × -9(-8, -9)

Therefore, the negative pair factors of 72 are

(-1, -72), (-2, -36) (-3, 24), (-4, 18), (-6, 12), and (-8, 9).

Prime Factorization of 72

72 is a composite number. Now let’s determine its prime factorization.

Factors of 72 by prime factorization
  • First, divide 72 by the smallest prime factor, which is 2.

72 ÷ 2 = 36

Once again, divide 36 by 2.

36 ÷ 2 = 18

18 ÷ 2 = 9

 When we divide 9 by 2, we get a fractional number, which cannot be a factor.

  • Let’s move on to the next prime number, which is 3.

9 ÷ 3 = 3

3 ÷ 3 = 1

  • As a result of the division process, we received a 1 and we can no longer proceed. 
  • Therefore, the prime factorization of 72 is 2 * 2 * 2 * 3 * 3 or 23 * 32, where 2 and 3 are prime numbers.
Links Related to Factors
Factors of 15= 1, 3, 5, and 15.Factors of 36=1, 2, 3, 4, 6, 9, 12, 18, and 36.
Factors of 48=1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.Factors of 18=1, 2, 3, 6, 9 and 18.
Factors of 42=1, 2, 3, 6, 7, 14, 21, and 42.Factors of 60=1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.
Factors of 35=1, 5, 7 and 35.Factors of 24=1, 2, 3, 4, 6, 8, 12, and 24.
Factors of 84= 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84.Factors of 50=1, 2, 5, 10, 25, 50.

Solved Examples

Example 1:

Find the common factors between 72 and 71.

Solution:

Among the factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

71 has two factors, 1 and 71.

Thus, 72 and 71 have a common factor of 1.

Example 2:

Calculate the common factors of 72 and 73.

Solution:

Factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.

Factors of 73 = 1 and 73.

The common factor of 72 and 73 is 1, since 73 is a prime number.

Example 3:

Calculate the common factor of 72 and 70.

Solution:

Factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.

Factors of 70 = 1, 2, 5, 7, 10, 14, 35, 70

Thus, the common factors of 70 and 72 are 1 and 2.

Practice Questions

  1. Find the common factor between 72 and 36.
  2. 72 and 144 have a common factor. What is it?
  3. How much is the sum of the factors of 72?
  4. What are the prime factors of 72?
  5. Calculate the common factors of 72 and 30.

FAQ`s

What are the factors of 72?

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72 are the factors of 72.

What is the prime factorization of 72?

72 has a prime factorization of 2 × 2 × 2 × 3 × 3 or 23 × 32.

What are the positive pair factors of 72?

Positive pair factors of 72 = (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), and (8, 9).

What are the negative pair factors of 72?

Negative pair factors of 72 = (-1, -72), (-2, -36), (-3, -24), (-4, -18), (-6, -12), and (-8, -9).

Is 24 a factor of 72?

It is true that 24 is a factor of 72. Since the number 24 divides 72 exactly without leaving any remainder, 24 is a factor of 72.

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