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Find the HCF and LCM

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Find the HCF and LCM

When finding the HCF and LCM of relatively low/easy numbers we can use the methods that we are used to such as find the HCF and LCM of 12 and 18

HCF

1218

1 x 121 x 18

2 x 62 x 9

3 x 43 x 6

Factors are:

1,2,3,4,6 and 12 1,2,3,6,9 and 18

Common factors are:

1,2,3,6

HCF is 6

LCM

List all the multiples of both numbers

12,24,36,48,60,72,84,96,108…

18,36,54,72,90,108…

Common multiples are 36,72,108 …..

LCM is 36

With harder numbers such as 36 and 90 we would literally be here all day working them out as these are larger numbers that we are not really used to so we have an easier method for these types of numbers. The other way is still correct but it just takes too long

Step 1

Split the numbers up to a product of prime numbers e.g.

36

Because 3 x 12 is equal to 36. We would circle or highlight the 3 as this is a prime number but 12 is not so we would split that again into 3 x 4. the three is a prime so we would circle or highlight that as it can not be split up further, finally we can split the 4 into 2 x 2 leaving the number 36 as a product of prime numbers

36 = 3 x 3 x 2 x 2 or 3² x 2²

No matter which way we split it we will always end up with the same answer as a product of prime numbers… see the next slide.

3

12

3

4

2

2

36

36

6

6

2

18

3

3

2

2

2

9

3

3

36 = 3 x 3 x 2 x 2 or 3² x 2²

So no matter which way we do the working out we will always end up with the same answer.

We could work out the product of prime numbers for 90 in the same way.

90 = 2 x 5 x 3 x 3

We can check the calculation by first seeing if they are all prime which they are and finally checking that these numbers do multiply to give 90… which they do

90

2

45

5

9

3

3

Step 2 – Put the product of prime numbers in a Venn diagram

90 = 2 x 5 x 3 x 3

36 = 2 x 2 x 3 x 3

36

90

2

5

2

3

3

These are the prime numbers that is extra in the product of 36

These are the prime numbers that is extra in the product of 90

These are the prime numbers that both numbers share so they go in the middle

36

90

2

5

2

3

3

In order to find the HCF we multiply everything in the middle section.

2 x 3 x 3 = 18 which is the HCF

In order to find the LCM we multiply all the numbers in the Venn diagram together

5 x 2 x 3 x 3 x 2 = 180 which is the LCM

Try this method with an even more difficult question

What is the HCF and LCM of:

36 and 48 (try the Venn diagram method although you can probably do it without)

45 and 70

110 and 125