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Lowest Common Multiples
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Lowest Common Multiples

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Lowest Common Multiples

The lowest common multiple is the smallest number that is a multiple of two other numbers.

1

A multiple is a number that is a product of two numbers. Which number below is a multiple of 55?

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2

So 135135 is a multiple of 55 because 5×27=1355\times 27=135.

It's also true that 135=15×9135=15\times 9.

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3

This means that 135135 is a multiple of 55 and 1515. What is the term for this?

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1

Sometimes we want to find the lowest common multiple of two numbers.

This is the smallest number that is a common multiple of them both. There are two ways we can approach this.

2

Let's find the lowest common multiple (LCM for short) of 88 and 66.

List the first three multiples of 88 separating your answers with a comma.

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3

Are any of these also a multiple of 66 and if so, which one?

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4

We have found the lowest common multiple of 66 and 88.

The LCM is 2424.

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1

Let's find the LCM of 1212 and 1515.

We list the multiples of the larger number - this means we have fewer multiples to find!

2

What are the first three multiples of 1515? Separate your answers with a comma.

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3

The first three multiples of 1515 are 1515, 3030 and 4545. Are any of these also a multiple of 1212?

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4

We need to find more multiples of 1515. The first three are 1515, 3030 and 4545. What are the next three? Separate your answers with a comma.

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5

The next three multiples of 1515 are 6060, 7575 and 9090. Are any of these also a multiple of 1212 and if so which one?

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6

We have found the LCM of 1212 and 1515 which is 6060.

Sometimes we have to find quite a few multiples, there is another method which can be quicker.

What is the LCM of 1212 and 1515 using prime factorisation?

1

This is the prime factorisation of 1515

33 and 55 are both prime numbers that multiply to make1515.

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2

What is the prime factorisation of 1212?

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3

We have the prime factorisation of both our numbers.

To find the LCM we eliminate the duplicate of any common prime factors the multiply the numbers that remain

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4

Which prime factors do each of these numbers have in common?

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5

We eliminate one of these common factors so that we don't duplicate it. Are there any other common prime factors?

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6

To find the LCM, we take the remaining prime factors and multiply them all together. What is the answer?

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7

We have used both methods to find the LCM of 1212 and 1515 is 6060

Prime factorisation means we don't need to find lots of multiples.

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1

Let's work through this one - find the LCM of 4040 and 9090.

The prime factorisation of 9090 is 2×32×52\times 3^2\times 5.

2

What is the prime factorisation of 4040?

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3

We now have the prime factorstions of both numbers. What are the common prime factors? Separate your answer with a comma.

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4

This time we need to be careful which prime factor we cross through.

Where a factor has a power, we cross through the factor with the lowest power.

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5

Find the LCM of 4040 and 9090 by multiplying out the remaining prime factors.

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6

Well done - you've found the LCM of 4040 and 9090!

The LCM is 360360.

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1

Let's find the LCM of 1616 and 4040. First find the prime factorisation of 1616.

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2

Now find the prime factorisation of 4040.

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3

The prime factorisations are: 16=2416=2^4

40=23×540=2^3\times 5 Which is the only common prime factor?

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4

The prime factorisations are: 16=2416=2^4

40=23×540=2^3\times 5 Taking out the duplicate factors, what is the LCM of 1616 and 4040?

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Use either method to find the Lowest Common Multiple of 1414 and 88.

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What is the LCM of 7575 and 9090

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Summary!

1

A multiple is a number that is a product of two numbers.

2020 is a multiple of 55 because 5×4=205\times 4=20.

2

A common multiple is a number that is a multiple of two other numbers.

2020 is a common multiple of 55 and 1010.

3

There are two ways of finding the LCM of two numbers

  1. List the multiples of the larger number and stop when you find a multiple of the smaller number.

  2. Use the prime factorisation technique

4
  1. List multiples of 1515 to find the LCM of 66 and 1515

Find multiples of 1515 until you reach a multiple of 66. These are 1515, 3030 and we stop there because 3030 is also a multiple of 66.

5
  1. Prime factorisation can also be used to find the LCM.

The LCM of 1515 and 2727 is 135135.

6

The prime factorisation of: 15=3×515=3\times 5

27=3327=3^3

Cross through the common prime factors, where there is a power, cross through the lowest power. This leaves 3×5{\cancel{3}}\times 5 and 333^3 giving the LCM as 33×5=1353^3\times 5=135.