# how many numbers between 1 to 72, do not have any common factor with 72?

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## Factors of 72 (Pair Factors and Prime Factors of 72)

The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72. Visit BYJU’S to learn the pair factors and the prime factors of 72 using the prime factorization method and many examples.

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## Factors Of 72

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 23 × 32**, where 2 and 3 are the prime numbers.

## Video Lesson on Prime Factors

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## Solved 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.

Learn more about factors and prime factors here with us in BYJU’S and also download BYJU’S – The Learning App for a better experience.

## Factors of 72

What are the Factors of 72? - Important Notes, How to Calculate Factors of 72 using Prime Factorization. Factors of 72 in Pairs, FAQs, Tips and Tricks, Solved Examples, and more.

## 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.

## 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 = 23 × 32; 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

## [Solved] How many even number of factors of number 72?

Prime factorization of 72 = 2 × 2 × 2 × 3 × 3 = 23 × 32 Total number of even factors of 2a × p1b × p2c × .. = a

Home Quantitative Aptitude Number System Integers

Question

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### How many even number of factors of number 72?

9 8 7 6

Answer (Detailed Solution Below)

Option 1 : 9 Detailed Solution

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Prime factorization of 72 = 2 × 2 × 2 × 3 × 3 = 23 × 32

Total number of even factors of 2a × p1b × p2c × .. = a × (b + 1) × (c + 1) [Where p1 and p2 are prime factor].

Hence, required number of even factors of 72 = 3 × (2 + 1) = 9

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Guys, does anyone know the answer?