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Lowest Common Multiples

# Lowest Common Multiples

### 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 $5$?

2

So $135$ is a multiple of $5$ because $5\times 27=135$.

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

3

This means that $135$ is a multiple of $5$ and $15$. What is the term for this?

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 $8$ and $6$.

List the first three multiples of $8$ separating your answers with a comma.

3

Are any of these also a multiple of $6$ and if so, which one?

4

We have found the lowest common multiple of $6$ and $8$.

The LCM is $24$.

1

Let's find the LCM of $12$ and $15$.

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

2

What are the first three multiples of $15$? Separate your answers with a comma.

3

The first three multiples of $15$ are $15$, $30$ and $45$. Are any of these also a multiple of $12$?

4

We need to find more multiples of $15$. The first three are $15$, $30$ and $45$. What are the next three? Separate your answers with a comma.

5

The next three multiples of $15$ are $60$, $75$ and $90$. Are any of these also a multiple of $12$ and if so which one?

6

We have found the LCM of $12$ and $15$ which is $60$.

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

What is the LCM of $12$ and $15$ using prime factorisation?

1

This is the prime factorisation of $15$

$3$ and $5$ are both prime numbers that multiply to make$15$.

2

What is the prime factorisation of $12$?

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

4

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

5

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

6

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

7

We have used both methods to find the LCM of $12$ and $15$ is $60$

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

1

Let's work through this one - find the LCM of $40$ and $90$.

The prime factorisation of $90$ is $2\times 3^2\times 5$.

2

What is the prime factorisation of $40$?

3

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

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.

5

Find the LCM of $40$ and $90$ by multiplying out the remaining prime factors.

6

Well done - you've found the LCM of $40$ and $90$!

The LCM is $360$.

1

Let's find the LCM of $16$ and $40$. First find the prime factorisation of $16$.

2

Now find the prime factorisation of $40$.

3

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

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

4

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

$40=2^3\times 5$ Taking out the duplicate factors, what is the LCM of $16$ and $40$?

Use either method to find the Lowest Common Multiple of $14$ and $8$.

What is the LCM of $75$ and $90$

Summary!

1

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

$20$ is a multiple of $5$ because $5\times 4=20$.

2

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

$20$ is a common multiple of $5$ and $10$.

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 $15$ to find the LCM of $6$ and $15$

Find multiples of $15$ until you reach a multiple of $6$. These are $15$, $30$ and we stop there because $30$ is also a multiple of $6$.

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

The LCM of $15$ and $27$ is $135$.

6

The prime factorisation of: $15=3\times 5$

$27=3^3$

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