In prime factorization, we factorize the numbers into prime numbers, called prime factors.

There are two methods of prime factorization:

Factorization means writing a given number as the product of two or more factors.

Prime factorization of a number is a way of showing a number as the product of prime numbers.

There are 2 methods to find the prime factors.

1. Division Method

2. Factor Tree Method

Prime Factorization by Division Method

Observe the following steps.

**I:** First we divide the number by the smallest prime number which divides the number exactly.

**II:** We divide the quotient again by the smallest or the next smallest prime number if it is not exactly divisible by the smallest prime number. We repeat the process again and again till the quotient becomes 1. Remember, we use only prime numbers to divide.

**III:** We multiply all the prime factors. Remember, the product is the number itself.

**Let us consider a few examples using division method.**

**1. Find the prime factors of 15.**

First Step: 2 is the smallest prime number. But it cannot divide 15 exactly. So, consider 3.

Second Step: Now, 5 cannot be divide by 3. Consider the next smallest prime number 5.

The prime factors of 15 are 3 × 5.

**2. Find the prime factors of 18.**

First Step: Consider 2, the smallest prime number.

Second Step: As 9 cannot be divide by 2. Consider the next smallest prime 3. Repeat the process till quotient becomes 1.

The prime factors of 18 are 2 × 3 × 3.

Prime factorization by factor tree method

Observe the following steps.

Suppose, we have to find the prime factors of 16

**1.** We consider the number 16 as the root of the tree.

**2.** We write a pair of factors as the branches of the tree
i.e., 2 × 8 = 16

**3.** We further factorize the composite factor 8 as 4 and 2,
and again the composite factors 4 as 2 and 2.

We repeat the process again till we get the prime factors of all the composite factors.

2 × 8 = 16

2 × 4 × 2 = 16

2 × 2 × 2 × 2 = 16

The prime factors of 16 = 2 × 2 × 2 × 2.

We can express the factor tree to find the prime factors of 16 in another way also.

4 × 4

2 × 2 × 2 × 2

The prime factors of 16 = 2 × 2 × 2 × 2.

Now, we will use both the methods (Factor Tree Method and Division Method) to the prime factorization of a number:

**1. The prime factorization of 36**

**Factor Tree Method**

Continue factorizing until only prime number remains.

36 = 2 × 2 × 3 × 3

**Division Method**

**Continue diving unit the quotient is a prime.**

**2. The prime factorization of 64.**

**Factor Tree Method**

Continue factorizing until only prime number remains.

64 = 2 × 2 × 2 × 2 × 2 × 2

**Division Method **

**Continue diving unit the quotient is a prime.**

**Questions and Answers on Methods of Prime Factorization:**

**I. Prime factorize the following numbers using division method.**

(i) 18

(ii) 125

(iii) 512

(iv) 144

(v) 360

(vi) 256

(vii) 96

(viii) 80

(ix) 625

(x) 169

**Answer:**

**I. **(i) 2 × 3 × 3

(ii) 5 × 5 × 5

(iii) 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

(iv) 2 x 2 x 2 x 2 x 3 x 3

(v) 2 × 2 × 2 × 3 × 3 × 5

(vi) 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

(vii) 2 × 2 × 2 × 2 × 2 × 3

(viii) 2 × 2 × 2 × 2 × 5

(ix) 5 × 5 × 5 × 5

(x) 13 × 13

**II. Find the prime factors using the factor tree.**

(i) 66

(ii) 75

(iii) 24

(iv) 156

(v) 128

**Answer:**

**II. **(i) 2 × 3 × 11

(ii) 3 × 5 × 5

(iii) 2 × 2 × 2 × 3

(iv) 2 × 2 × 3 × 13

(v) 2 × 2 × 2 × 2 × 2 × 2 × 2

**III. Copy and complete these factor trees.**

**Answer:**

**III. **(i) 2 × 2 × 5 × 5

(ii) 2 × 2 × 2 × 2 × 3

(iii) 5 × 19

(iv) 3 × 3 × 7

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