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Factor Tree Method

Breaking down a number into its prime factors by constructing a tree like structure is called factor
tree method and the structure is called the factor tree.

In this method, we follow these steps:
Step 1: First break down the number into two factors.

Step 2: Check whether the two factors are prime or composite. If it is a composite number, break
down it again as done in step 1. The two factors of a number are like branches of the factor tree.
Step 3: Repeat the above steps until all branches end with prime numbers.
Example 10: Find the prime factorisation of the following numbers using the factor tree method.

(a) 60 (b) 56
Solution: We can make the factor trees for the given numbers as follows:

(a) 60 or 60 (b) 56 or 56

2 30 6 10 7 8 4 14

3 10 2 3 2 5 2 4 2 2 2 7

2 5 2 2

\ 60 = 2 × 2 × 3 × 5 \ 56 = 2 × 2 × 2 × 7
Prime Factorisation to Check if Two Numbers are Co-prime

In order to check whether two numbers are co-prime or not, prime factorisation method can be
used. After prime factorisation, we will check, if there is any prime factor that is common in the
prime factorisation of both the numbers or not.

Let us consider the numbers 40 and 135.
To check if they are co-prime, we can use the prime factorisation of both numbers.
40 = 2 × 2 × 2 × 5 and 135 = 3 × 3 × 3 × 5.

Now, we see that 5 is a prime factor of 40 as well as 135. Therefore, 40 and 135 are not co-prime.
What about 42 and 55? Their prime factorisations are as follows:

42 = 2 × 3 × 7 and 55 = 5 × 11.
There are no common prime factors. So, we conclude that they are co-prime.
Therefore, we can say that if there are no common prime factors, then the two numbers are co-
prime.

Example 11: State whether 80 and 231 are co-prime.
Solution: The prime factorisations of 80 and 231 are as follows:
80 = 2 × 2 × 2 × 2 × 5 and 231 = 3 × 7 × 11

We see that there are no common prime factors. Indeed, the prime factors of 80 are 2 and 5 while,
the prime factors of 231 are 3, 7, and 11. Therefore, 80 and 231 are co-prime.

151 Prime Time