Factors of 72 are the numbers, which gives the result as 72 when multiplied together in a pair of two. For example, 2×3 = 6, states that 2 and 3 are the factors of 6. Basically, multiples of 72 give an extended timetable of 72, such as 72, 144, 216, 288, 360, 432, 504, 576, 648 and so on. To find the factors of a number, 72, we will use the factorization method. The factors of 72 can be represented either in positive or negative forms. But the factors of 72 cannot be a decimal or fraction. For example, the factors of 72 can be (1, 72) or (-1, -72). If we multiply a pair of a negative number, such as multiplying -1 and -72, it will result in the original number 72.
In this article, we are going to learn the factors of 72, and the pair factors and the prime factors of 72 using the prime factorization method with many solved examples.
Table of Contents:
- What are the Factors of 72?
- Pair Factors of 72
- Prime Factorization of 72
- Solved Examples
- Practice Questions
- FAQs
What are the Factors of 72?
The factors of 72 are the numbers that divide 72 exactly without leaving any remainder. In other words, the factors of 72 are the numbers that are multiplied in pairs resulting in an original number 72. As the number 72 is a composite number, it has many factors other than one and the number itself. Thus 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 2^{3} × 3^{2} |
Pair Factors of 72
The pair factors of 72 are the pair of numbers, which are multiplied together resulting in an original number 72. The pair factors of 72 can be positive pair or negative pair. If we multiply the pair of negative numbers, it will result in an original number 72. Thus, the positive and the negative pair factors of 72 are as follows:
Positive Pair Factors of 72:
Positive Factors of 72 |
Positive 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) |
Hence, the positive pair factors of 72 are (1, 72), (2, 36), (3, 24), (4, 18), (6, 12) and (8, 9).
Negative Pair Factors of 72:
Negative Factors of 72 |
Negative 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) |
Hence, the negative pair factors of 72 are (-1, -72), (-2, -36), (-3, -24), (-4, -18), (-6, -12) and (-8, -9).
Prime Factorization of 72
The number 72 is a composite number. Now let us find the prime factorisation of this number.
- The first step is to divide the number 72 with the smallest prime factor,i.e. 2.
72 ÷ 2 = 36
Again, divide 36 by 2.
36 ÷ 2 = 18
18 ÷ 2 = 9
Now, if we divide 9 by 2 we get a fraction number, which cannot be a factor.
- Now, proceed to the next prime numbers, i.e. 3.
9 ÷ 3 = 3
3 ÷ 3 = 1
- We have received 1 at the end of the division process and further, we cannot proceed. So, the prime factorisation of 72 is 2 × 2 × 2 × 3 × 3 or 2^{3} × 3^{2}, where 2 and 3 are the prime numbers.
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Examples
Example 1:
Find the common factors of 72 and 71.
Solution:
The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
The factors of 71 are 1 and 71.
Thus, the common factor of 72 and 71 is 1.
Example 2:
Find 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.
As 73 is a prime number, the common factor of 72 and 73 is 1.
Example 3:
Find the common factors of 72 and 70.
Solution:
The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
The factors of 70 are 1, 2, 5, 7, 10, 14, 35, 70
Therefore, the common factors of 70 and 72 are 1 and 2.
Practice Questions
- Find the common factors of 72 and 36.
- What is the common factor of 72 and 144?
- What is the sum of all the factors of 72?
- What are the prime factors of 72?
- Find the common factors of 72 and 30.
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Frequently Asked Questions on Factors of 72
What are the factors of 72?
The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
What is the prime factorization of 72?
The prime factorization of 72 is 2 × 2 × 2 × 3 × 3 or 2^{3} × 3^{2}.
What are the positive pair factors of 72?
The positive pair factors of 72 are (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), and (8, 9).
What are the negative pair factors of 72?
The negative pair factors of 72 are (-1, -72), (-2, -36), (-3, -24), (-4, -18), (-6, -12), and (-8, -9).
Is 24 a factor of 72?
Yes, 24 is a factor of 72. As the number 24 divides 72 exactly without leaving any remainder, 24 is a factor of 72.