Lowest common multiple and highest common factor

In a nutshell

The lowest common multiple and highest common factor of two numbers is an application of prime factorisation and Venn diagrams.

The lowest common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.

1.	Find the prime factorisation of both numbers.
2.	Draw a Venn diagram, with each circle representing a number.
3.	If a prime factor is in both numbers, put the prime factor in the middle part of the Venn diagram (where both circles intersect) once (not twice). If a prime factor is in only one of the numbers, put that prime factor in the correct circle.
4.	The lowest common multiple is found by multiplying all the prime factors together.

What is the lowest common multiple of $16$ and $18$ ?

First, find the prime factorisation of both numbers using the factor tree method:

$16=2\times2\times2\times2$

$18=2\times3\times3$

Draw a Venn diagram and put any common prime factors in the middle:

$2$ is in both numbers only once, so write down $2$ in the middle of the diagram.

For the $16$ circle, write down the remaining $2$ s (three of them).

For the $18$ circle, write down the two $3$ s.

Multiply all the numbers together to obtain the LCM:

$2\times2\times2\times2\times3\times3=144$

The lowest common multiple of $\underline{16}$ and $\underline{18}$ is $\underline{144.}$ 

The highest common factor (HCF) of two numbers is the largest number that is a factor of both numbers.

1.	Find the prime factorisation of both numbers
2.	Draw a Venn diagram, with each circle representing a number
3.	If a prime factor is in both numbers, put the prime factor in the middle part of the Venn diagram (where both circles intersect) once (not twice). If a prime factor is in only one of the numbers, put that prime factor in the correct circle.
4.	The highest common factor is found by multiplying all the prime factors that are only in the middle part of the Venn diagram together.

What is the highest common factor of $16$ and $18$ ?

The prime factorisation and Venn diagram has already been done in the above example.

The HCF is found by multiplying all the numbers in the middle - which is just $2$ .

The highest common factor of $\underline{16}$ and $\underline{18}$ is $\underline{2.}$ 

The HCF and LCM of two numbers, $a$ and $b$ , follow the formula:

$HCF\times LCM=a\times b$

The HCF and LCM of $16$ and $18$ are $2$ and $144$ respectively.

$HCF\times LCM=2\times144=\underline{288}$

$16\times18=\underline{288}$

Hence, the formula works in this case.

Note: You don't need to know this formula, but it may be helpful to use to check if your answer is correct.