Objective

This topic will teaches us about 

1. Prime Factorisation
2. Highest Common Factor(HCF)
3. Lowest Common Multiple(LCM)

Prime Factorisation

Before learning about prime factorization, let's have a look at prime numbers. Prime number is a number that is exactly divisible by only 1 and itself. The first few prime numbers are: 2, 3, 5, 7, 11, 13, and 17 ..., etc. Prime factorization is the process of finding the prime number multiplying which we can get a certain number.

Find the prime factors of 8: 8 = 2 * 2* 2.

Here, 2 is the prime factor of 8.  A number that doesn't have any prime factors except itself is a prime number. An exact divisor of a number is called a factor. Factors are the numbers you multiply together to get another number.

There are different methods which can be utilized to find the prime factorization of a number. One way is by the division method as below:

Another method is the Factor Tree. It is a diagram used to break down a number by dividing it by its factors until all the numbers left are prime. 

Highest Common Factor

 The Highest Common Factor (HCF) of two or more given numbers is the highest (or greatest) of their common factors. It is also known as Greatest Common-divisor (GCD).

The HCF of 20, 28 and 36 can also be found by prime factorisation of these numbers as follows:

Thus, 20 = 2 × 2 × 5;

           28 = 2 × 2 × 7;

           36 = 2 × 2 × 3 × 3;

The common factor of 20, 28 and 36 is 2(occurring twice). Thus, HCF of 20, 28 and 36 is  2 × 2 = 4.

If the highest common factor (HCF) of two numbers is equal to 1, then they are called co-prime or relatively prime. HCF of co-prime numbers 4 and 15 was found as follows by factorisation:

4 = 2 × 2  and  15 = 3 × 5;

Since there is no common prime factor, so HCF of 4 and 15 is 1. 

Lowest Common Multiple

The Lowest Common Multiple (LCM) of two or more given numbers is the lowest (or smallest or least) of their common multiples.

Consider the following example. 

The prime factorisations of 12 and 18 are:

12 = 2 × 2 × 3;

18 = 2 × 3 × 3;

In these prime factorisations, the maximum number of times the prime factor 2 occurs is two; this happens for 12. Similarly, the maximum number of times the factor 3 occurs is two; this happens for 18. The LCM of the two numbers is the product of the prime factors counted the maximum number of times they occur in any of the numbers.

Thus, in this case LCM = 2 × 2 × 3 × 3 = 36

The other method is to find the LCM of the given numbers using the division method:

So, LCM = 2 × 2 × 3 × 5 × 5;

(A)  Divide by the least prime number which divides atleast one of the given numbers. Here, it is 2. The numbers like 25 are not divisible by 2 so they are written as such in the next row.

(B)  Again divide by 2. Continue this till we have no multiples of 2.

(C)  Divide by next prime number which is 3.

(D)  Divide by next prime number which is 5.

(E)  Again divide by 5.

Summary 

1. Prime factorization is the process of breaking up a number in to factors until only prime factors are left.
2. The advantage of Factor Tree is that we can get the factors in order.
3. We have also seen that,
   
   • The Highest Common Factor (HCF) of two or more given numbers is the highest of their common factors.
   • The Lowest Common Multiple (LCM) of two or more given numbers is the lowest of their common multiples.