**Prime Factorisation**

**Prime Factorisation**. *It is the process of finding the (prime) factors, which are multiplied together to get the original number* (*order in which the factors appear does not matter*).

For example, we can write

- 24 = 2 × 2 × 2 × 3
- 24 = 2 × 2 × 3 × 2
- 24 = 2 × 3 × 2 × 2
- 24 = 3 × 2 × 2 × 2

and the same prime factors (there are three 2’s and one 3) occur in all the cases.

##### More divisibility rules

- Product of two consecutive whole numbers is divisible by 2
- If a number is divisible by another number, then it is divisible by each of the factors of that number.
- If a number is divisible by two co-prime numbers, then it is divisible by their product also.
- If two given numbers are divisible by a number, then their sum is also divisible by that number.
- If two given numbers are divisible by a number, then their difference is also divisible by that number.

** Example**. Find the prime factorisation of 324.

Prime factorization of 324

= 2 × 2 × 3 × 3 × 3 × 3.

** Example**. Find the prime factorisation of 400.

Prime factorization of 400

= 2 × 2 × 2 × 2 × 5 × 5.

** Example**. Find the prime factorisation of 20570.

Prime factorization of 20570 = 5 × 2 × 11 × 11 × 17.

** Example**. Find the prime factorisation of 58500.

Prime factorization of 58500 = 5 × 5 × 5 × 3 × 3 × 2 × 2 × 13.

** Example**. Find the prime factorisation of 45470971.

Prime factorization of 45470971 = 13 × 13 × 7 × 7 × 17 × 17 × 19.

**Factor Tree**. *A factor tree is a diagram used to determine the prime factors of a natural number greater than one.*

The number 30 can be written as 6 × 5. Again, 6 can then be written as 2 × 3. Definitely, the number 30 can be factored different ways, but the result is always has one 2, one 3 and one 5. Below we will show a few factor tree for the number 30.

** Example**. Find the factor tree of 48.

** Example**. Find the factor tree of 60.

**Exercise 1**. Which of the following statements are true? Justify.

- (a) If two numbers are co-primes, at least one of them must be prime.
- (b) If a number is divisible by 3, it must be divisible by 9.
- (c) All numbers which are divisible by 4 must also be divisible by 8.
- (d) If a number exactly divides two numbers separately, it must exactly divide their sum.
- (e) If a number is divisible by 9, it must be divisible by 3.
- (f) All numbers which are divisible by 8 must also be divisible by 4.
- (g) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.
- (h) A number is divisible by 18, if it is divisible by both 3 and 6.
- (i) If a number is divisible by 9 and 10 both, then it must be divisible by 90.

**Exercise 2**. Which factors are not included in the prime factorisation of a composite number? Justify.

**Exercise 3**. Write the greatest four-digit number and express it in terms of its prime factors.

**Exercise 4**. Write the smallest five-digit number and express it in the form of its prime factors.

**Exercise 5**. Find all the prime factors of 1729 and arrange them in ascending order. Now state the relation, if any; between two consecutive prime factors.

**Exercise 6**. The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples.

**Exercise 7**. The sum of two consecutive odd numbers is divisible by 4. Verify this statement with the help of some examples.

**Exercise 8**. In which of the following expressions, prime factorisation has been done?

- (
**a**) 24 = 2 × 3 × 4 - (
**b**) 54 = 2 × 3 × 9 - (
**c**) 56 = 7 × 2 × 2 × 2 - (
**d**) 70 = 2 × 5 × 7

**Exercise 9**. I am the smallest number, having four different prime factors. Can you find me?