# Common Factors and Common Multiples

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Observe the factors of some numbers taken in pairs.

What are the factors of 4 and 18?

The factors of 4 are 1,

4 | 1, | 2, | 4 |

The factors of 18 are 1, 2, 3,

18 | 1, | 2, | 3, | 6, | 9, | 18 |

Let us see both the factors together.

4 | = | 1 | × | 2 | × | 4 | |||||||||

18 | = | 1 | × | 2 | × | 3 | × | 6 | × | 9 | × | 18 | |||

----- | ----- | ----- | ----- | ----- | ----- | ----- | ----- | ||||||||

(Common for 4 and 18) | = | 1 | × | 2 |

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What are the common factors of 4 and 15?

4 | = | 1 | × | 2 | × | 4 | |||||

15 | = | 1 | × | 3 | × | 5 | × | 15 | |||

----- | ----- | ----- | ----- | ----- | ----- | ||||||

(Common for 4 and 15) | = | 1 |

These two numbers have only

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**(a) 8, 20**

**Find the common factors of 8 and 20**

8 | = | 1 | × | 2 | | × | 4 | |× | 8 | ||||||||

20 | = | 1 | × | 2 | × | 4 | × | 5 | × | 10 | × | 20 | ||||

----- | ----- | ----- | ----- | ----- | ----- | ----- | ----- | ----- | ||||||||

(Common for 8 and 20) | = | 1 | 2 | 4 |

These two numbers have only

**(b) 9, 15**

9 | = | 1 | × | 3 | | × | 9 | ||||

15 | = | 1 | × | 3 | × | 5 | × | 15 | ||

----- | ----- | ----- | ----- | ----- | ----- | |||||

(Common for 9 and 15) | = | 1 | 3 |

These two numbers have only

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What about 7 and 16?Common factor is

**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?

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,

Factors of 16 are 1, 2, 4,

4 | = | 1 | × | 2 | × | 4 | ||||||||||||

12 | = | 1 | × | 2 | × | 3 | × | 4 | × | 6 | × | 12 | ||||||

16 | = | 1 | × | 2 | × | 4 | × | 8 | × | 16 | ||||||||

----- | ----- | ----- | ----- | ----- | ----- | |||||||||||||

(Common for 4,12 and 16) | = | 1 | × | 2 | × | 4 |

Clearly,

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**Highest Common Factor**

**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

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**

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**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?

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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 **Greatest Common Factor** or **gcf** of 18 and 30.

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Once again, we can use the

18 | = | 2 | × | 3 | × | 3 | ||

30 | = | 2 | × | 3 | × | 5 |

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.

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

**X** = 2 × 3 = 6.

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

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

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

**prime factors.**

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## 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,

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

We observe that 12, 24, 36, **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,

We say that lowest common multiple of 4 and 6 is

It is the smallest number that both the numbers are

**LCM of 20, 25 and 30 using Division method.**

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We write the numbers as follows in a row :

2 | 20 | 25 | 30 | ||

2 | 10 | 25 | 15 | ||

3 | 5 | 25 | 15 | ||

5 | 5 | 25 | 5 | ||

5 | 1 | 5 | 1 | ||

1 | 1 | 1 |

**2**. The numbers like 25 are not divisible by 2 so they are written as such in the next row.

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**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?

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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 **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

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**.

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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):

12 | 60 | |

2 × 2 × 3 | 2 × 2 × 3 × 5 |

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

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To find the lcm of **40** and **60**, we first need to find the

40 | = | 2 | × | 2 | × | 2 | × | 5 | ||

60 | = | 2 | × | 2 | × | 3 | × | 5 |

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**.

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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**).

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

- Find the prime factorisation of each number.
- 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**:

12 | = | 2 | × | 2 | × | 3 | ||||

30 | = | 2 | × | 3 | × | 5 | ||||

45 | = | 3 | × | 3 | × | 5 |

Therefore the lcm of **12**, **30** and **45** is 2 ×

Prime numbers are a special case: the lcm of two different primes is simply their

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**Find the LCM of 20, 25 and 30.**

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

20 | = | 2 | × | 2 | × | 5 | ||||

25 | = | 5 | × | 5 | ||||||

30 | = | 2 | × | 3 | × | 5 |

Therefore the lcm of **20**, **25** and **30** is 2 ×