A number in general has two diverging aspects. A number can be viewed as a factor of a larger number and at the same it is formed as product of some smaller numbers.The concepts of factor decomposition and product composition are applied at any level of Mathematical problem solving.

Prime Factorization of a number is finding the prime factors of the number. Every natural number greater than 1, is either a prime or a product of primes in a unique way.

Prime Factorization means breaking up a composite number into prime factors. The concept of factors and the idea of factorization originate from the divisibility of numbers. If a number divides another number with a remainder "0", then the second number is said to be Prime Factorization of a number is finding the prime factors of the number. Every natural number greater than 1, is either a prime or a product of primes in a unique way.

**divisible**by the first number.

### Solved Example

**Question:**Find the prime factors of 12.

**Solution:**

When 12 is divided by 3, the quotient is 4 and the remainder is 0. So we can say that 12 is divisible by 3.

Hence prime factors of 12,

12 = 2 x 2 x 3 = 2

Hence prime factors of 12,

12 = 2 x 2 x 3 = 2

^{2}x 3.### Solved Example

**Question:**Prime factorization of 18.

**Solution:**

Prime factors of 18,

18 = 2 x 3 x 3 = 2 x 3

or 3 x 3 x 2 = 3

or 3 x 2 x 3 = 3

18 = 2 x 3 x 3 = 2 x 3

^{2}or 3 x 3 x 2 = 3

^{2}x 2or 3 x 2 x 3 = 3

^{2}x 2.Divisible by |
Divisibility Test |

2 |
The ones digit is 0, 2, 4, 6 or 8. In other words the number should be an even number. |

3 |
The sum of the digits is divisible by 3. Example: 231 231 = 2 + 3 + 1 = 6, 6 is divisible by 3. So 231 is divisible by 3. |

5 |
The ones digit is 0 or 5 Example: 120 Here ones digit is zero So 120 is divisible by 5. |

7 |
Double the ones digit and subtract that from the sum of the rest of the digits.Repeat the procedure till a two digit number is arrived. If the two digit number got is divisible by 7, then the given number is divisible by 7. |

11 |
Subtract the sum of the digits in odd places from the sum of the digits in the even places. If the difference obtained is 0 or divisible by 11, then the given number is divisible by 11. |

### Solved Example

**Question:**Write -

### Prime Factorization of 72

**Solution:**

By divisibility test:

Since 72 is an even number, is divisible by 2

72 = 2 x 36.

Continuing the division with 2,

36 = 2 x 18 and 18 = 2 x 9.

9 = 3 x 3

Hence prime factorization of 72,

Since 72 is an even number, is divisible by 2

72 = 2 x 36.

Continuing the division with 2,

36 = 2 x 18 and 18 = 2 x 9.

9 = 3 x 3

Hence prime factorization of 72,

**72 = 2 x 2 x 2 x 3 x 3 = 2**^{3}x 3^{2}**factor tree**. The factor tree is formed by branching out the factors at each level of prime division.

### Solved Example

**Question:**Use factor tree to find -

### Prime Factorization of 245

**Solution:**

The ones digit is 5, so the number is divisible by 5.

Dividing 245 by 5 we get the quotient as 49.

again 49 is divisible by 7 and 49 = 7 x 7.

The factor tree can be extended and completed as follows:

The process is stopped at that level when the quotient on division by a prime is also a prime.

Hence the prime factorization is 245,

Dividing 245 by 5 we get the quotient as 49.

again 49 is divisible by 7 and 49 = 7 x 7.

The factor tree can be extended and completed as follows:

The process is stopped at that level when the quotient on division by a prime is also a prime.

Hence the prime factorization is 245,

**245 = 5 x 7 x 7 = 5 x 7**^{2}.### Solved Example

**Question:**Find the greatest common factor of the numbers 24 and 60.

**Solution:**

The Prime factorization of the two numbers 24 and 60 is shown here::

Prime factors of 24 =

Prime factors of 60 =

Hence

Prime factors of 24 =

**2**x**2**x 2 x**3**Prime factors of 60 =

**2**x**2**x**3**x 5Hence

**GCF of 24 and 60 = 2 x 2 x 3 = 12**### Solved Examples

**Question 1:**Write -

### Prime Factorization of 36

**Solution:**

By divisibility test:

36 is even number, a number with 6 as ones digit.

So 2 is a prime factor of 36. On division by 2 the quotient is 18.

=> 36 = 2 x 18

Again 18 is an even number and can be written as 18 = 2 x 9.

Now we have 36 = 2 x 2 x 9, 9 is divisible by 3 and 9 = 3 x 3

Hence the prime factorization of

If the repeated factors are replaced by powers, then

36 is even number, a number with 6 as ones digit.

So 2 is a prime factor of 36. On division by 2 the quotient is 18.

=> 36 = 2 x 18

Again 18 is an even number and can be written as 18 = 2 x 9.

Now we have 36 = 2 x 2 x 9, 9 is divisible by 3 and 9 = 3 x 3

Hence the prime factorization of

**36,**

3636

**= 2 x 2 x 3 x 3**.If the repeated factors are replaced by powers, then

**36 = 2**^{2}x 3^{2}.**Question 2:**Write -

### Prime Factorization of 48

**Solution:**

By divisibility test:

48 being an even number is divisible by 2.

48 = 2 x 24

Continuing the division with 2,

24 = 2 x 12 and 12 = 2 x 6 and 6 = 2 x 3.

Hence the prime factorization for 48 is

48 being an even number is divisible by 2.

48 = 2 x 24

Continuing the division with 2,

24 = 2 x 12 and 12 = 2 x 6 and 6 = 2 x 3.

Hence the prime factorization for 48 is

**48 = 2 x 2 x 2 x 2 x 3 = 2**^{4}x 3.**Question 3:**Write -

### Prime Factorization of 75

**Solution:**

By divisibility test:

As the ones digits is 5, this number is divisible by 5.

Dividing 75 by 5 the quotient is 15, 75 = 5 x 15.

15 can be factored in a similar manner as 15 = 3 x 5.

Hence the prime factorization of 75 is

As the ones digits is 5, this number is divisible by 5.

Dividing 75 by 5 the quotient is 15, 75 = 5 x 15.

15 can be factored in a similar manner as 15 = 3 x 5.

Hence the prime factorization of 75 is

**75 = 5 x 5 x 3 = 5**^{2}x 3