Page 147 - Ganit Kaushal

(I) Breakdown into prime factors: We break down the number into its prime factors as follows.

Examples: The prime factorisation of 24 = 4 × 6 = 2 × 2 × 2 × 3.
The prime factorisation of 42 = 6 × 7 = 2 × 3 × 7.

\ The prime factors of 24 are 2 (repeated thrice) and 3, and the prime factors of 42 are 2, 3, and 7.
(II) Factor Tree Method: The other method for prime factorisation of a number 24
is by using the Factor Tree.
Let us find the prime factorisation of 24 using a factor tree. 6 4

Think of a factor pair 6 × 4 = 24. So, one set of factors for 24 is 6 and 4.
Continue to factor; we can break down 6 to 2 × 3, and 4 can be broken down 2 3 2 2
to 2 × 2. All the factors are now prime, so the factorisation of 24 = 2 × 2 × 2 × 3.

Similarly, the prime factorisation of 90 is shown Similarly, the prime factorisation of 135 is
using a factor tree. shown by a factor tree.

Prime factorisation of 90 = 2 × 3 × 3 × 5. Prime factorisation of 135 = 3 × 3 × 3 × 5.
90 135

15 6 45 3

3 5 2 3 9 5

3 3

(III) Division Method:
Here, we divide the number by the least possible prime and get the quotient. The quotient is further
divided by the least prime, and we continue this process till a prime is left. We give here a few
examples to understand the process of the division method.

Example 1. Find the prime factorisation of Example 2. Find the prime factorisation of
315 by the division method. 2904 by the division method.

3 315 Æ prime 3 is a factor of 315 2 2904 Æ prime 2 is a factor of 2904
3 105 Æ prime 3 is a factor of 105 2 1452 Æ prime 2 is a factor of 1452
5 35 Æ prime 5 is a factor of 35 2 726 Æ prime 2 is a factor of 726
7 Æ stop here as 7 is prime 3 363 Æ prime 3 is a factor of 363
11 121 Æ prime 11 is a factor of 121
\ 315 = 3 × 3 × 5 × 7. 11 Æ stop here as 11 is prime

\ 2904 = 2 × 2 × 2 × 3 × 11 × 11.
Example 3. Find the prime factorisation of 13915 by the division method.

5 13915 Æ prime 5 is a factor of 13915
11 2783 Æ prime 11 is a factor of 2783
11 253 Æ again 11 is a factor of 253
23 Æ stop here as 23 is prime

\ 13915 = 5 × 11 × 11 × 23.

Prime Time 145