**Lcm and HCF**

## To understand this Method first, You need to know what is Prime Factorisation ...

The

is the largest whole number which is a factor of both.

**highest common factor (HCF)**of two whole numbersis the largest whole number which is a factor of both.

**HCF Example**

**Consider the numbers 12 and 15:**

The factors of 12 are : 1, 2, 3, 4, 6, 12.

The factors of 15 are : 1, 3, 5, 15.

1 and 3 are the only common factors (numbers which are factors of

Therefore, the highest common factor of 12 and 15 is 3.

The factors of 12 are : 1, 2, 3, 4, 6, 12.

The factors of 15 are : 1, 3, 5, 15.

1 and 3 are the only common factors (numbers which are factors of

__both__12 and 15).Therefore, the highest common factor of 12 and 15 is 3.

The

is the smallest whole number which is a multiple of both.

**lowest common multiple (LCM)**of two whole numbersis the smallest whole number which is a multiple of both.

**LCM Example**

Consider the numbers 12 and 15 again:

The multiples of 12 are :

**12, 24, 36, 48, 60, 72, 84, ...**.

The multiples of 15 are :

**15, 30, 45, 60, 75, 90, ...**.

**60**is a

**common multiple**(a multiple of

__both__12 and 15), and there are no lower common multiples.

Therefore, the

**lowest common multiple**of 12 and 15 is

**60**.

## Although the methods above work well for small numbers, they are

more difficult to follow with bigger numbers. Another way to find the

highest common factor and lowest common multiple of a pair of

numbers is to use the prime factorisations of the two numbers.

**Finding HCF & LCM with prime factorisations**

We want to find the HCF and LCM of the numbers 60 and 72.Start by writing each number as a product of its prime factors.

**60 = 2 * 2 * 3 * 5**

**72 = 2 * 2 * 2 * 3 * 3**

To make the next stage easier, we need to write these so that each new prime factor begins in the same place:

**60**

**= 2**

*** 2**

*** 3**

*** 5**

**72**

**= 2**

*** 2**

*** 2**

*** 3**

*** 3**

All the "2"s are now above each other, as are the "3"s etc. This allows us to match up the prime factors.

The highest common factor is found by multiplying all the factors which appear in

__both__lists:

So the HCF of 60 and 72 is

The lowest common multiple is found by multiplying all the factors which appear in

**2 × 2 × 3**which is**12**.The lowest common multiple is found by multiplying all the factors which appear in

__either__list:So the LCM of 60 and 72 is

**2 × 2 × 2 × 3 × 3 × 5**which is**360**.