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Lowest common multiple and highest common factor

Lowest common multiple and highest common factor

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Tutor: Bilal

Summary

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.



Lowest common multiple (LCM)

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


Finding the lowest common multiple of two numbers

​​procedure

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.


Example 1

What is the lowest common multiple of 1616 and 1818?


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

16=2×2×2×216=2\times2\times2\times2​​

18=2×3×318=2\times3\times3​​


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

22is in both numbers only once, so write down 22 in the middle of the diagram.

For the 1616 circle, write down the remaining 22s (three of them).

For the 1818 circle, write down the two 33s.


Maths; Types of numbers; KS3 Year 7; Lowest common multiple and highest common factor


Multiply all the numbers together to obtain the LCM:

2×2×2×2×3×3=1442\times2\times2\times2\times3\times3=144​​


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



Highest common factor (HCF)

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


Finding the highest common factor of two numbers

procedure

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.


Example 2

What is the highest common factor of 1616 and 1818?


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


Maths; Types of numbers; KS3 Year 7; Lowest common multiple and highest common factor


The HCF is found by multiplying all the numbers in the middle - which is just 22.


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



A helpful formula

The HCF and LCM of two numbers, aa​ and bb, follow the formula:

HCF×LCM=a×bHCF\times LCM=a\times b


Example 3

The HCF and LCM of 1616 and 1818 are 22 and 144144 respectively.


HCF×LCM=2×144=288HCF\times LCM=2\times144=\underline{288}​​

16×18=28816\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.


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