Common Factors and Common Multiples

Observe the factors of some numbers taken in pairs.

What are the factors of 4 and 18?

The factors of 4 are 1, and 4.

The factors of 18 are 1, 2, 3, , and 18.

These two numbers have only 12 as the common factor.

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(a) 8, 20

Find the common factors of 8 and 20

These two numbers have only 32 as the common factor.

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(b) 9, 15

These two numbers have only 23 as the common factor.

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What about 7 and 16?Common factor is 17

Two numbers having only 1 as a common factor are called co-prime numbers.

Thus, 4 and 15 are co-prime numbers.

Are 7 and 15, 12 and 49, 18 and 23 co-prime numbers? yesno

Can we find the common factors of 4, 12 and 16?

Factors of 4 are 1, 2 and .

Factors of 12 are 1, 2, 3, 4, and .

Factors of 16 are 1, 2, 4, and .

Clearly, , and are the common factors of 4, 12, and 16.

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Highest Common Factor

We can find the common factors of any two numbers. We now try to find the highest of these common factors.

What are the common factors of 12 and 16? They are 1, 2 and .

What is the highest of these common factors? It is .

What are the common factors of 20, 28 and 36? They are 1, 2 and 45 and again 4 is highestlowest of these common factors.

But why should we find the HCF. What kind of problems can we solve by knowing how to calculate the HCF. Let's explore.

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Find the HCF of 24 and 36

Factors of 24: ,,,,,,,

Factors of 36: ,,,,,,,,

The common factors of 24 and 36 are: ,,,,,

The greatest common factor is .

Uses of HCF in real life

An architect is planning the floor for a large courtyard that measures 18m by 30m. She wants it to be covered in quadratic tiles, without any gaps or overlaps along the sides. What is the largest size of squares she can use?

Just like before, this question is not about geometry - it is about divisibility. The length of the sides of the tiles has to divide both 18 and 30, and the largest possible number with that property is 67. This is called the Greatest Common Factor or gcf of 18 and 30.

Once again, we can use the prime factorisation to calculate the gcf of any two numbers. Remember that any factor of a number must have some of the prime factors of that number.

Suppose that X is the gcf of 18 and 30. Then X divides 18 so the prime factors of X must be among 2, 3 and 3. Also, X divides 30 so the prime factors of X must be among 2, 3 and 5.

To find X, we simply need to multiply all numbers which are prime factors of bothone of 18 and 30:

X  =  2 × 3  =  6.

Now we have a simple method for finding the gcf of two numbers:

1. Find the prime factorisation of each number.
2. Multiply the prime factors which are in both numbers.

Once again prime numbers are special: the gcf of two different primes is always 13, because they don’t share any prime factors.

LCM

Let us now look at the multiples of more than one number taken at a time.

What are the multiples of 4 and 6?

The multiples of 4 are 4, 8, 12, , , , , , , ...

The multiples of 6 are 6, 12, 18, , , , , , , ... (write a few more)

Out of these, are there any numbers which occur in both the lists?

We observe that 12, 24, 36, , are multiples of both 4 and 6.

Can you write a few more?

They are called the common multiples of 4 and 6.

What are the common multiples of 4 and 6? They are 12, 2436, 3612, 4836,... . Which is the lowest of these? It is 1236.

We say that lowest common multiple of 4 and 6 is .

It is the smallest number that both the numbers are factorsmultiples of this number.

LCM of 20, 25 and 30 using Division method.

We write the numbers as follows in a row :

Uses of LCM in real life

Two runners are training on a circular racing track. The first runner takes 60 seconds for one lap. The second runner only takes 40 seconds for one lap. If both leave at the same time from the start line, when will they meet again at the start?

This question really isn’t about the geometry of the race track, or about velocity and speed – it is about multiples and divisibility.

The first runner crosses the start line after 60 seconds, 120 seconds, 180 seconds, 240 seconds, and so on. These are simply the multiplesfactors of 60. The second runner crosses the start line after 40 s, 80 s, 120 s, 160 s, and so on. The first time both runners are back at the start line is after seconds.

What we’ve just found is the smallest number which is both a multiple of 40 and a multiple of 60. This is called the lowest common multiple or lcm.

We saw how to find LCM using long division method above. Now we will see how to use prime factor method to find LCM. To find the LCM of any two numbers, it is important to realise that if a divides b, then b needs to have all the prime factors of a (plus some more):

This is easy to verify: if a prime factor divides a, and a divides b, then that prime factor must also divide b.

To find the lcm of 40 and 60, we first need to find the prime factorisation of both:

Suppose that X is the lcm of 40 and 60. Then 40 divides X, so 2, 2, 2 and 5 must be prime factors of X. Also, 60 divides X, so 2, 2, 3 and 5 must be prime factors of X.

To find X, we simply combine all the prime factors of 40 and 60, but any duplicates we only need once:

X  =  2 × 2 × 2 × 3 × 5

This gives us that X = 120, just like we saw above. Notice that if the same prime factor appears multiple times, like 2 above, we need to keep the maximum occurrences in one of the two numbers (3 times in 40 is more than 2 times in 60).

Now we have a simple method for finding the lcm of two numbers:

1. Find the prime factorisation of each number.
2. Combine all prime factors, but only count duplicates once.

We can use the same method to find the lcm of three or more numbers at once, like 12, 30 and 45:

Therefore the lcm of 12, 30 and 45 is 2 × × 3 × 3 × = 180.

Prime numbers are a special case: the lcm of two different primes is simply their productsumdifference, because they don’t have any common prime factors which would get “canceled”.

Find the LCM of 20, 25 and 30.

Find the lcm of three or more numbers at once, like 20, 25 and 30:

Therefore the lcm of 20, 25 and 30 is 2 × × × 5 × = 300.