We know that factors of a number divide that particular number completely. Let us talk about the number 36. There are a total of 9 factors of 36; hence it is a composite number. There are 2 prime numbers among the 9 factors of 36 which are termed as prime factors of 36. In this lesson, we will calculate the factors of 36, prime factors of 36, and factors of 36 in pairs along with solved examples.

**Factors of 36:**1, 2, 3, 4, 6, 9, 12, 18, and 36.**Prime Factorization of 36:**2 × 2 × 3 × 3 = 2² × 3²

Let us explore more about factors of 36 and ways to find them.

**Table of Contents**

- What Are the Factors of 36?
- How to Calculate the Factors of 36?
- Factors of 36 in Pairs
- Important Notes
- FAQs on Factors of 36
- Factors of 36 Solved Examples
- Challenging Questions
- Interactive Questions

## What Are the Factors of 36?

Factors of a number are the numbers that divide the given number exactly without any remainder. According to the definition of factors, factors of 36 are those numbers that exactly divide 36 leaving no remainder. In other words, we can say if two numbers are multiplied and the product is 36, then the numbers are factors of 36.

Let us do a hit and trial method to find factors of 36.

- Divide 36 ÷ 2 = 18
- Quotient = 18, remainder = 0
- Now multiply 18 × 2 = 36
- Hence, 18 and 2 are the factors of 18.
- 36 is a composite number as it has factors other than 1 and itself.

## How to Calculate the Factors of 36?

We can use different methods like divisibility test, prime factorization, and the upside-down division method to calculate the factors of 36.

The most common method we use to find factors is prime factorization method.

- In prime factorization, we express 36 as a product of its prime factors.
- In the division method, we see which numbers divide 36 exactly without a remainder.

Let us calculate factors of 36 using the follwoing two methods:

- Factors of 36 by prime factorization factor tree method
- Factors of 36 by upside-down division method

### Prime Factorization of 36 by Upside-Down Division Method

Now as discussed above, in prime factorization method we show factors of 36 as a product of prime factors of 36. Follow the steps to find the prime factors of 36 by prime factorization method.

- With the help of divisibility rules, find out the smallest factor of the given number. Here, 36 is an even number. So it is clearly divisible by 2. In other words, 2 divides 36 with no remainder. Therefore, 2 is the smallest prime factor of 36.
- 36 ÷ 2 = 18. Now find the prime factors of the obtained quotient. Repeat Step 1 and Step 2 till we get a prime number as the quotient. Here, 18 is the quotient.
- 18 ÷ 2 = 9. Here, 9 is the quotient. Now find the prime factors of the 9.
- 9 ÷ 3 = 3. Here 3 is the prime number. So we can stop the process.
- Prime factorization of 36 using upside-down division method is 2 × 2 × 3 × 3 = 2² × 3².

### Prime Factorization of 36 by Factor Tree Method

Factor tree method is another visualization method to represent factors of 36. It also helps in determining prime factors of 36. A factor tree is not unique for a given number. Instead of expressing 36 as 2 × 18, we can express 36 as 6 × 6.

- Prime factorization of 36 using factor tree method is 2 × 2 × 3 × 3 = 2² × 3².

**Explore factors using illustrations and interactive examples**

- Factors of 33: The factors of 33 are 1, 3, 11, and 33.
- Factors of 34: The factors of 34 are 1, 2, 17, and 34.
- Factors of 38: The factors of 38 are 1, 2, 19, and 38.
- Factors of 30: The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.
- Factors of 360: The factors of 360 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360.
- Factors of 35: The factors of 35 are 1, 5, 7, and 35.
- Factors of 37: The factors of 37 are 1 and 37.

## Factors of 36 in Pairs

Factor pairs are the two numbers which, when multiplied, give the number 36. Find all of the factors of the number 36 using the steps below:

- Step I: Start with 1 and the number itself.
- Step II: Count up by ones to see if you can multiply two numbers together to get your target number.
- Step III: Stop when you can’t get any more numbers in between.
- Step IV: Connect the factor pairs.

Factor pairs of 36 are given below:

- 36 = 1 × 36
- 36 = 4 × 9
- 36 = 6 × 6
- 36 =12 × 3
- 36 = 18 × 2

Let us club the numbers in pairs.

- The factors of 36 in pairs are (1, 36), (4, 9), (6, 6), (12, 3), and (18, 2).
- There are total 5 pairs of factors of 36.

**We can have negative factors also for a given number.**

- Since the product of two negative numbers is positive [(-) × (-) = +].
- The negative pair factors of 36 are (-1, -36), (-4, -9), (-6, -6), (-12, -3), and (-18, -2)
- There are total 5 negative pairs of factors of 36.

**Important Notes:**

- 36 is a composite number as it has factors other than 1 and itself.
- Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
- Pair factors of 36 are (1, 36), (4, 9), (6, 6), (12, 3), and (18, 2).
- 1 is a factor of every number.
- Prime factorization of 36 is 2 × 2 × 3 × 3 = 2² × 3².

## FAQs on Factors of 36

### What are the prime factors of 36?

The prime factors of 36 are 2 and 3.

### Is 36 a square number?

A perfect square is a number that can be expressed as the product of two equal integers.

As 36 can be expressed as the product of two equal numbers, 36 is a square number.

### Which is the smallest factor of 36?

1 is the smallest factor of 36. 2 is the smallest prime factor of 36.

### Which is the highest factor of 36?

36 is the highest factor of 36. Every number is the highest factor of itself.

### What are the common factors of 36 and 6?

Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Factors of 6 are 1, 2, 3, and 6.

The common factors of 36 and 6 are 1, 2, 3, and 6.

## Solved Examples

**Example 1: **Help Ria in listing out the factors of 36 and its factor pairs.

**Solution:**

- 36 = 1 × 36
- 36 = 4 × 9
- 36 = 6 × 6
- 36 =12 × 3
- 36 = 18 × 2

Therefore, factors of 36 are : 1, 2, 3, 4, 6, 9, 12, 18, and 36.

The factors of 36 in pairs are (1, 36), (4, 9), (6, 6), (12, 3), and (18, 2).

**Example 2: **Find the average of the sum of factors of 36.

**Solution:**

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Sum of all the factors of 36 is = 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 + 36 = 91

The average of the sum of factors of 36 is = 91 ÷ 9 = 10.11

**Example 3: **Twelve friends have 36 cookies from a shop and distributed among themselves equally. How many cookies would each one of them get?

**Solution:**

36 cookies are to be divided among 12 friends equally.

This implies we need to divide 36 by 12.

36 ÷ 12 = 3

Hence, each friend will get 3 cookies.

**Challenging Questions:**

- Find the factors of 3600 with the help of techniques used in finding factors of 36.
- Prove that the factor of a number is always less than or equal to the number itself?

## Interactive Questions

**Here are a few activities for you to practice.**

**Select/Type your answer and click the "Check Answer" button to see the result.**