Factors of 72: Pair factors and Prime Factors – Mindspark

What are The Factors of 72?

Some important facts about the factors of the number 72 are given below.

• Factors of the number 72 are the numbers that divide it wholly and leave zero as a remainder. So, we get a whole number as the quotient.
• When we multiply any two whole numbers with each other, and the result is 72, we can say that both the numbers are the factors of 72.
• 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72 are the factors of the number 72.
• The pair factors can be expressed either in positive or negative forms. But these can not be decimal or fraction. For example, the pair factors of the number 72 can be (2, 36) or (-2, -36)

How To Calculate the Factors?

Factors of 72 are the numbers that divide it wholly without leaving any remainder. Every number is completely divisible by 1 and the number itself. So, the two factors will be 1 and the number itself, which is 72. 

To find other factors, we will do the divisibility test.

Using the divisibility rules from 1 to 10, we find that 72 is divisible by 2, 3, 4, 6, 8, and 9. By dividing 72 with these numbers, we can find other factors.

So, divide 72 with 2; we get (72 ÷ 2) = 36 as the quotient and zero as the remainder.

Since the quotient is 36 and the remainder is 0. So, 72 will be completely divisible by quotient 36 also. Therefore, 36 is also a factor of 72.

Similarly, when we divide 72 by 3, 4, 6, and 8, we get 24, 18, 12 and 9, respectively. All these numbers are the required factors.

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

Pair Factors of 72

To find the pair factors, we will have to find pairs of such numbers whose product is 72. These pairs are – 

1 × 72 = 72

2 × 36 = 72

3 × 24 = 72

4 × 18 = 72

6 × 12 = 72

8 × 9 = 72

Hence, the positive pair factors are (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), and (8, 9).

Similarly, the negative pair factors will be (-1, -72), (-2, -36), (-3, -24), (-4, -18), (-6, -12), and (-8, -9). It is because the product of two negative numbers is a positive number.

Prime Factorization of 72

To find the prime factors, we will have to divide 72 by prime numbers till the quotient obtained is 1. Let’s understand how. 

The smallest prime factor of 72 is 2(because 72 is an even number). So divide 72 by 2, and we get;

            72/2 = 36

The smallest prime factor of 36 is also 2. Now divide 36 by 2, and we get;

            36/2 = 18

Similarly, divide 18 by 2, and we get 9. 

Then divide 9 by 3, and we get 3.

Finally, divide 3 by 3, and we get 1.

Now, we can’t divide 1 by any further by prime factors. 

Therefore, prime factorization of 72 is equal to (2 x 2 x 2 x 3 x 3) or \left(2^{3} \times 3^{2}\right).

Examples

1. What is the sum of all the factors of the number 72?

Solution:

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

So, the sum of these factors = 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 36 + 72 = 195

Therefore, 195 is the required answer.

2. Find the factors of 72 among the numbers given below:

     16, 18, 20, 22, 24

Solution:

18 and 24 are the factors of 72 among the given numbers since these two numbers divide 72 wholly without leaving any remainder.