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Factors of 36: Prime Factorization, Methods, Tree, and Examples

Factors of 36 refer to the numbers by which 36 can completely be divided. It means that these are the numbers that when divided by 36 leave no remainder. Thus, a factor is a number that can divide with other numbers evenly

An extremely easy way to check factors of a specific number is by first listing all the numbers less than or equal to the number you’re finding factors of. For example, in the case of 36, the numbers will be from 1 to 18.

You then have to divide each of them to find the answer. A fun fact about factors is that 1 is the factor of all numbers! However, there are two ways of finding factors of a number that are division and multiplication methods. 

Though, there isn’t just one way to find factors of integers. There is a trick to finding factors of a number in an even more simplified way which is that you just have to keep dividing the number and when there’s such a case where the remainder becomes 0, you consider the quotient and divisor both as a factor of the specific number.

Let’s take an example of such a case.

If you divide the number 36 by 2 it will give you a conclusion that both the divisor 2 and the answer 18 will be factors of 36 and they also form a factor pair. They are considered as its factors because the remainder is zero and the quotient is 36. 

\[ 2 \times 36 = 18 \]

In this article, you’ll get a quick go-through of details of the factors of 36. This article consists of details of trouble-free solutions on how to find and determine the factors of 36, fun facts you might not know about them as well as examples and solutions to the factors of 36.

What Are the Factors of 36?

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The number 36 has 9 factors as it is a composite number. 

Each of these factors can be paired into factor pairs. It can be done by pairing the numbers that give 36 as the product. The remainder will always be zero when 36 is divided by these numbers.

How To Calculate the Factors of 36?

You can calculate the factors of 36 in more than a single way such as by using the division method. Let’s see how you can find out the factors of 36 by using the techniques mentioned at the start of this article. 

First, jot down half of your given number i.e. the half of 36 is 18. This means that you will check for the divisibility of 36 from numbers starting from 1 to 18. 

Keep in mind the fact that to become a factor of 36 the number it is being divided with has to give a remainder zero and the divisors should only produce whole number quotients. If the number gives off an answer in decimal, it will not be considered a factor either. 

To get a more clear vision of this concept, let’s look at the division of 36 into two numbers that are 2 and 5.

\[ \frac{36}{2} = 18 \]

\[ \frac{36}{5} =  7.2\]

Since a whole number quotient is obtained only from the division of 36 from 2 out of both these numbers, 2 is a factor of 36. 

In addition to this as it also has no remainder, hence not just 2 but the quotient of such a divisor is also a factor. Thus, both 2 and 18 are factors of 36.

All the possible divisions of 36 are mentioned below:

\[ \frac{36}{1} = 36\]

\[ \frac{36}{2} = 18 \]

\[ \frac{36}{3} = 12 \]

\[ \frac{36}{4} = 9 \]

\[ \frac{36}{6} = 6 \]

All the above-mentioned divisions produce zero as the remainder so possible factors of 36 are:

Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36

Factors of 36 by Prime Factorization

Prime factorization is a good way of figuring out which prime factors can multiply with each other to give the number as a product or it can be defined as a way of expressing a specific number as a product of its prime factors. 

Moreover, a prime number is a number that has 2 factors only – 1 and the number itself. 

So, to get the prime factor of 36 you have to keep breaking down the quotient by division until the number 1 is received. The method is more like taking the least common multiple of a number.

However, the only difference is that prime factorization is the product of prime numbers that are equal to the actual number.

For the number 36, you can choose to divide 36 by 2 and 3 as they are its prime numbers. You can find the prime number by the way mentioned below:

\[ \frac{36}{2} = 18 \]

You have to continue the same process till you get 1 as the quotient. 

\[ \frac{18}{2} = 9 \]

\[ \frac{9}{3} = 3 \]

\[ \frac{3}{3} = 1 \]

Therefore, the prime factors of 36 are the numbers 2 and 3. 

The prime factorization of 36 is also shown below:

Figure 1

We can also write this prime factorization mathematically as written below;

\[ 2^{2} \times 3^{2} = 36 \]

Factor Tree of 36

There isn’t just one way even to represent the factors of a number. Expressing factors through a Factor Tree is one of the many ways to visually represent the prime factors of a specific number.

The factor tree starts with the number itself and the branches extend representing factors till you get the prime number on the tree.

According to the prime factorization, 2 and 3 are the prime factors of the number 36. Thus 3 should be the last number represented on the factor tree.

The Factor Tree of 36 is shown below:

Figure 2

Some unique and interesting facts about the number 36 are given below:

  1. 36 is a square triangular number since it is the square of 6 and a triangular number. The only other triangular number whose square root is likewise a triangular number is this one, making it the smallest square triangular number other than 1. 
  2. Furthermore, the integer 36 is the product of the squares of the first three integers, 1, 2., and 3. The specific sum of the cubes of the first three integers, and the sum of a twin prime as well.
  3. Not just this, each tip of a standard pentagram has an internal angle of 36 degrees. Even the atomic number of the element krypton is 36 in the periodic table.
  4. Another fun fact is that both the digits 3 and 6 are multiples of three and if we add up both the digits i.e. 3+6 it gives the answer 9, which is also a multiple of 3.

Factors of 36 in Pairs

Factor Pairs are a set of two integers that give the number itself as an answer when multiplied together. Let’s take the same case as an example. It refers to the two numbers, which when multiplied, will produce 36.

There are positive and negative Factor pairs as well, all you have to do is reverse the signs. 

The factor pairs of 36 are mentioned below: 

\[ 1 \times 36 = 36 \] 

\[ 2  \times 18 = 36 \]

\[ 3  \times 12 = 36 \]

\[ 4  \times 9 =  36 \]

\[ 6  \times 6 = 36 \]

Thus there are 5-factor pairs of the number 36 which are (1,36), (2, 18), (3, 12), (4,9), and (6, 6).

Factors of 36 Solved Examples

To further clarify how to determine the factors of 36 and how to evaluate them, some solved examples are given below. 

Example 1

How many specific odd numbers are there in the factors of the number 36?

Solution

You first have to take an overview of all the factors of 36 which are:

Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36 

Thus by looking at the factors it can be easily determined that 36 has 3 odd numbers as factors, which are given below:

Odd factors of 36: 1, 3, 9 

Example 2

What are the positive and negative factor pairs of 36 and how can we find them?

Solution

We can find factor pairs by the multiplication of two numbers that give an answer which is equal to the product i.e. 36. 

Thus the possible combinations can be e.g.

 \[ 1 \times 36 = 36  \]

 \[2 \times 18 = 36  \]

 \[3 \times 12 = 36 \] 

And a few more. To get the negative factor pairs all you have to do is reverse the signs e.g. (2, 18) will become (-2, -18). 

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

The negative pair factors of 36 are (-1,-36), (-2, -18), (-3, -12), (-4, -9), and (-6, -6).

Example 3

What are the even factors of the number 36?

Solution

Even numbers are the numbers divisible by two and the ones that can be divided into two equal groups. Thus to find the even factors of 36, you will first need to find out all the factors of 36 and then list down all the ones that are divisible by 2. 

The factors of 36 are written below:

Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36 

Thus, the even factors of the number 36 are:

Even factors: 2, 4, 6, 12, 18, 36

Images/mathematical drawings are made by using GeoGebra.

Factors of 35 | Factor List | Factors of 37

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