How do you write 36 as a product of its prime factors?

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Let’s first understand what are prime numbers and factors.

What is a factor?

A factor is a number that divides the given number evenly or completely. This means on dividing, the remainder will be zero.

For example, 3 and 2 are factors of 36 i.e. they divide 36 completely.

Also on dividing 36 by 2 or 3, the remainder turns out to be zero.

What are prime numbers?

Prime numbers are those numbers that have only two factors i.e. 1 and the number itself.

No other number is able to divide them completely.

For example, 3 is a prime number and just has two factors 1 and 3.

The numbers that are not prime are called composite numbers.

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36 and its factors

The number 36 is an even number i.e. it has 6 at its unit’s place. Every even number is composite.

So, let’s begin finding factors of 36.

Method 1: Division Method

We start by 2:

1. 36 on dividing by 2 gives 16

This means we can represent 36 as a multiplication of 18 and 2.

36 = 18 * 2 ……………………………………………………. Equation (1)

2. Let’s further break it. Here, 18 is again an even number s it ends with 8 at the unit’s place and 2 is a prime number as it has just two factors 1 and 2.

So, 18 can further be expressed into a product of two numbers i.e.

18 = 9 * 2

Replacing the value of 18 in equation (1), we get

36 = 9 * 2 * 2 ……………………………………………………. Equation (2)

3. Now we have got two 2’s which are prime and a 9 which we know is a composite number.

9 is obtained by multiplying two 3’s i.e. 9 can be expressed as a product of two 3’s.

9 = 3 * 3;

We can replace 9 in equation (2) and we get:

36 = 3 * 3 * 2 * 2

Now, all the numbers or factors of 36 are prime. No further breaking is required.

So, the 36 can be written as a product of two 3’s and 2’s or;

36 = 2^2 * 3^2

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Another way to find factors is the prime factorization method or upside division method.

Method 2: Prime factorization

Method:

With the help of divisibility rules, first, you need to find the smallest factor of the given number. Divide and write the quotient.
The process is repeated with every quotient obtained till the quotient comes out to be 1.

Let’s do the prime factorization of 36:

2	36
2	18
3	9
3	3
	1

Prime factorization is the best way to obtain a number as a product of all prime numbers.

So, by looking at the left side we can easily note that

36 = 2 * 2 * 3 * 3

Method 3: Factor tree method

The factor tree method is one of the most effective in grasping the concept of expressing a number into a product of prime numbers as it involves visualization.

Here we take the number in the node and the child nodes contain the numbers or factors of that number.

For that, we just need one relation or any two numbers that on multiplication give the desired number.

So, for 36 we can either go for 2 * 18 or we can also pick 6 * 6 or 9 * 4

For convenience we take 36 = 2 * 18.

Now, multiply all the last child nodes. This will give you the answer.

36 = 2 * 2 * 3 * 3 or 2^2 * 3^2

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