**Finding the prime factorization of a number :**

Prime factorization is the method of expressing a number as a product of prime numbers.

Factors are numbers we can multiply together to get a number.A multiple is the product of a number and any whole number except zero.

For example,

1, 2,3 and 6 are the factors of 6.

Prime factors of a number means, we have to choose only prime numbers among factors of the given number. So that their product will be the original number.

But 2 and 3 are the prime factors of 6.

Usually we have two methods of finding the prime factorization of a number

(i) Factor tree method

(ii) Using a ladder diagram

Using the steps given below, we can find the prime factors of a number.

**Step 1 :**

List the factor pairs of the given number.

**Step 2 :**

Choose any factor pair to begin the tree. If a number in this pair is prime, circle it. If a number in the pair can be written as a product of two factors, draw additional branches and write the factors.

**Step 3 : **

Continue adding branches until the factors at the ends of the branches are prime numbers.

**Step 4 : **

Write the prime factorization of the given number. We can use exponents to represent repeated factors.

The tricks given below will be helpful to find the prime number which exactly divides the given number.

- A number which ends with 0, 2, 4, 6 and 8 is divisible by the smallest prime number 2.
- A number which ends with 0 or 5 is divisible by 5
- If the sum of digits of the given number is a multiple of 3, then the given number is divisible by 3.

Let us see an example problem to understand the concept of finding the prime factorization of a number

**Example 1 : **

Use a factor tree to find the prime factorization of 240.

**Solution :**

The symbol · means “times.”

240 = 2 x 2 x 2 x 2 x 3 x 5

We can use exponents for repeated factors.

So, 240 = 2⁴ x 3 x 5

Hence, prime factors of 240 are 2⁴, 3 and 5.

A ladder diagram is another way to find the prime factorization of a number.

Steps followed in the above method :

**Step 1 : **

Put the given number inside the "L" shape

**Step 2 :**

We have to split the given number by prime numbers only. That is, always we have to put prime numbers out side the "L" shape.

**Step 3 :**

The tricks given below will be helpful to find the prime number which exactly divides the given number.

- A number which ends with 0, 2, 4, 6 and 8 is divisible by the smallest prime number 2.
- A number which ends with 0 or 5 is divisible by 5
- If the sum of digits of the given number is a multiple of 3, then the given number is divisible by 3.

**Step 4 :**

Take the first digit of the given number and check how many times the prime number goes in to that.

Further process is explained in the examples given below.

**Example 2 :**

Find the prime factors of 324 by using ladder diagram

**Solution :**

**Step 1 :**

Since the given number ends with 4, first we have to split the given number by the smallest even prime number 2.

**Step 2 :**

2 goes into 3 one time.We have 1 left. If we take this 1 along with the next digit 2, we get 12. If we divide this by 2, we get 6.

We don’t have any number remaining in 12. So we can take the next digit 4. Again, if we divide 4 by 2, we get 2.

**Step 2 :**

If we repeat this process, we will get

So, the prime factors of 324 = 2 x 2 x 3 x 3 x 3 x 3

Let us see the next example on "finding the prime factorization of a number"

**Example 3 :**

Using a factor tree and a ladder diagram to find the prime factorization of 54.

**Solution :**

**Factor tree method :**

**Ladder diagram method :**

Prime factors of 54 are 2 and 3³.

Note :

In both ways, we will get the same answer.

Let us see the next example on "finding the prime factorization of a number"

**Example 4 :**

Find the prime factors of 625 by using ladder diagram

**Solution :**

**Step 1 :**

Since the given number ends with 5, first we have to split it by the prime number 5.

**Step 2 :**

5 goes into 6 one time.We have 1 left. If we take this 1 along with the next digit 2, we get 12. Again we have to divide it by 5. If we divide this by 5, we get 2.

Now we have 2 left. Now we have to take this 2 along with the next digit 5, we get 25. If we divide 25 by 5, we get 5.

**Step 3 :**

By repeating this process until we get prime factors.

Hence, prime factors of 625 = 5⁴

Let us see the next example on "finding the prime factorization of a number"

**Example 5 :**

Find the prime factors of 4096 using ladder diagram

**Solution :**

Prime factors of 4096

= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

**Example 6 :**

Find the prime factors of 400 by using ladder diagram

**Solution :**

Prime factors of 400

= 2 x 2 x 2 x 2 x 5 x 5

**Example 7 :**

Find the prime factors of 144

**Solution :**

Prime factors of 144

= 2 x 2 x 2 x 2 x 3 x 3

**Example 8 :**

Find the prime factors of 1024

**Solution :**

Prime factors of 1024

= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

**Example 9 :**

Find the prime factors of 256

**Solution :**

Prime factors of 256

= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

**Example 10 :**

Find the prime factors of 2025

**Solution :**

Prime factors of 2025

= 5 x 5 x 3 x 3 x 3 x 3

After having gone through the stuff given above, we hope that the students would have understood "Finding the prime factorization of a number".

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