Factors of 72
Factors of 72 are those numbers that divide 72 completely without leaving any remainder. There are 12 factors of 72 among which 72 is the biggest factor and 2 and 3 are its prime factors. The prime factorization of 72 can be done by multiplying all its prime factors such that the product is 72. Let us learn about the factors of 72, the prime factorization of 72, and the factor tree of 72 in this article.
1.  What are the Factors of 72? 
2.  Prime Factorization of 72 
3.  Factor Tree of 72 
4.  Factors of 72 in Pairs 
5.  FAQs on Factors of 72 
What are the Factors of 72?
There are 12 factors of 72 that can be listed as 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. It means that 72 is completely divisible by all these numbers. Apart from these, 72 also has negative factors that can be listed as, 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72. For negative factors, we need to multiply a negative factor by a negative factor, like, (36) × (2) = 72.
How to Find the Factors of 72?
Factorization of a number means writing the number as a product of its factors. The most commonly used method to find the factors of a number is using the multiplication method. Let us find the factors of 72 using the multiplication method.
Factors of 72 using Multiplication
Let us find the factors of 72 using multiplication with the help of the following steps.
 Step 1: In order to find the factors of 72 using multiplication, we need to check what pairs of numbers multiply to get 72. So, we divide 72 by natural numbers starting from 1 and go on till 9. We need to make a note of those numbers that divide 72 completely.
 Step 2: The numbers that completely divide 72 are known as its factors. We write that particular number along with its pair and make a list as shown in the figure given above. As we check and list all the numbers up to 9, we automatically get the other pair factor along with it. For example, starting from 1, we write 1 × 72 = 72, and 2 × 36 = 72 and so on. Here, (1, 72) forms the first pair, (2, 36) forms the second pair and the list goes on as shown. So, as we write 1 as the factor of 72, we get the other factor as 72; and as we write 2 as the factor of 72, we get 36 as the other factor. Like this, we get all the factors.
 Step 3: After the list is noted, we get all the factors of 72 starting from 1 up there, coming down and then we go up again up to 72. This gives us a complete list of all the factors of 72 as shown in the figure given above.
Therefore, the factors of 72 can be listed as 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. Now, let us learn about the prime factorization of 72.
Prime Factorization of 72
Prime factorization is a way of expressing a number as a product of its prime factors. The prime factors of a number are those factors that are prime numbers. The prime factorization of 72 can be done using the following steps. Observe the figure given below to understand the prime factorization of 72.
 Step 1: The first step is to divide the number 72 with its smallest prime factor. We know that a prime factor is a prime number which is a factor of the given number. In this case, it is 2. So, 72 ÷ 2 = 36
 Step 2: We need to repeatedly divide the quotient by 2 until we get a number that is no more divisible by 2. So, we divide 36 again by 2 which is 36 ÷ 2 = 18
 Step 3: Divide 18 again by 2 which results in 18 ÷ 2 = 9
 Step 4: Now, 9 is not completely divisible by 2, so, we proceed with the next prime factor of 72, which is 3. That is 9 ÷ 3 = 3
 Step 5: Divide the quotient by 3 again which is 3 ÷ 3 = 1
 Step 6: We need not proceed further as we have obtained 1 as our quotient.
 Step 7: Therefore, the prime factorization of 72 is expressed as 2 × 2 × 2 × 3 × 3 = 2^{3} × 3^{2}; where 2 and 3 are prime numbers and the prime factors of 72.
Factor Tree of 72
We can also find the prime factors of 72 using a factor tree. The factor tree of 72 can be drawn by factorizing 72 until we reach its prime factors. These factors are split and written in the form of the branches of a tree. The final factors are circled and are considered to be the prime factors of the 72. Let us find the prime factors of 72 using the following steps and the factor tree given below.
 Step 1: Split 72 into two factors. Let us take 4 and 18.
 Step 2: Observe these factors to see if they are prime or not.
 Step 3: Since both 4 and 18 are composite numbers, they can be further split into more factors. Hence, we repeat the process of factorizing them and splitting them into branches until we reach the prime numbers.
 Step 4: Here, 4 can be further split into 2 and 2. Similarly, 18 can be further split into 2 and 9. Then 9 can be further split into 3 and 3. At this stage, we are left with prime numbers, 2 and 3. We circle them since we know that they cannot be factorized further. This is the end of the factor tree.
 Step 5: Therefore, the prime factors of 72 = 2 × 2 × 2 × 3 × 3
Note: It should be noted that there can be different factor trees of 72. For example, we can start by splitting 72 into 2 and 36. Then, 36 can be split further into 2 and 18. Then, 18 can be split further into 2 and 9. After this 9 can be split into 3 and 3. Finally, we can observe the same prime factors, that is, 72 = 2 × 2 × 2 × 3 × 3
Factors of 72 in Pairs
The factors of 72 can be written in pairs. This means that the product of the pair factors is 72. The factors of 72 in pairs can be written as shown in the table given below:
Factors  Positive Pair Factors 
1 × 72 = 72  1, 72 
2 × 36 = 72  2, 36 
3 × 24 = 72  3, 24 
4 × 18 = 72  4, 18 
6 × 12 = 72  6, 12 
8 × 9 = 72  8, 9 
It is possible to have negative pair factors as well because the product of two negative numbers also gives a positive number. Let us have a look at the negative pair factors of 72.
Factors  Negative Pair Factors 
1 × 72 = 72  1, 72 
2 × 36 = 72  2, 36 
3 × 24 = 72  3, 24 
4 × 18 = 72  4, 18 
6 × 12 = 72  6, 12 
8 × 9 = 72  8, 9 
The following points explain some features of the pair factors of 72.
 The pair factors of the number 72 are whole numbers in pairs that are multiplied to get the original number, i.e., 72.
 Pair factors could be either positive or negative but they cannot be fractions or decimal numbers.
 The positive pair factors of 72 are as follows: (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), and (8, 9). The negative pair factors of 72 are (1, 72) (2, 36) (3, 24)(4, 18)(6, 12) and (8, 9)
Important Notes
 Only whole numbers and integers can be the factors of a number.
 Only composite numbers can have more than two factors. Since 72 is a composite number, it has more than two factors.
 Every factor of a given number is either less than or equal to the given number.
 The number of factors of a given number is finite. 72 has 12 factors.
 Factors of 72 are those numbers that divide 72 completely without leaving any remainder.
 72 has a total of 12 factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.
 There is a trick to calculate the total number of factors of a number. For example, 72 = 2 × 2 × 2 × 3 × 3 = 2^{3 }× 3^{2}. We get the prime factorizations of 72 as 2^{3 }× 3^{2}. Just add one (1) to the exponents 3 and 2 individually and multiply their sums. (3 +1) × (2 +1) = 4 × 3 = 12. This means 72 has 12 factors in all.
Points to remember
Let us recollect the list of the factors, the negative factors, and the prime factors of 72.
 Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72
 Negative Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72
 Prime Factors of 72: 2, 3
 Prime Factorization of 72: 2 × 2 × 2 × 3 × 3 = 2^{3} × 3^{2}
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Examples on Factors of 72

Example 1: State true or false with respect to the factors of 72.
a.) 3 and 6 are factors of 72.
b.) 2 and 3 are the prime factors of 72.
Solution:
a.) True, 3 and 6 are factors of 72.
b.) True, 2 and 3 are the prime factors of 72.

Example 2: List the positive and negative pair factors of 72.
Solution:
The positive and negative pair factors of 72 are as follows: (1, 72), (1, 72), (2, 36), (2, 36), (3, 24), (3, 24), (4, 18), (4, 18), (6, 12), (6, 12), (8, 9), and (8, 9)

Example 3: List all the positive factors of 72.
Solution:
The positive factors of 72 can be listed as 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.
FAQs 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, 72 and its negative factors are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
What are the Prime Factors of 72?
2 and 3 are the two prime factors of 72. The prime factors of a number are those factors that are prime numbers. In this case, if we do the prime factorization of 72, we get 2 × 2 × 2 × 3 × 3,= 2^{3 }× 3^{2 }where 2 and 3 are prime numbers and the prime factors of 72.
What are the Composite Factors of 72?
There are 12 positive factors of 72. Out of these, the composite factors of 72 are 4, 6, 8, 9, 12, 18, 24, 36, and 72. The remaining factors, 2 and 3 are prime factors while 1 is neither a composite nor a prime number.
What is the Sum of the Factors of 72?
The sum of all the factors of 72 can be calculated by adding 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 which is 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 36 + 72 = 195.
How Many Factors of 72 are also common to the Factors of 19?
The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 and the factors of 19 are 1, 19. We can see that 72 and 19 have only one common factor which is 1. Therefore, 72 and 19 are coprime.
What are the Pair Factors of 72?
The positive pair factors of 72 can be listed as follows, (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), (8, 9). The negative pair factor factors can be listed as follows, (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), and (8, 9)
What is the Greatest Common Factor of 72 and 63?
The factors of 72 and 63 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 and 1, 3, 7, 9, 21, 63 respectively. The common factors of 72 and 63 are (1, 3, 9). Hence, the Greatest Common Factor (GCF) of 72 and 63 is 9.
What are the Common Factors of 72 and 98?
The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 and the factors of 98 are 1, 2, 7, 14, 49 and 98. We can see that their common factors are 1 and 2.
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