** MULTIPLES**

A multiple of a number X is the number obtained when X is multiplied by 1 or 2 or 3 or 4 or 5 and so on e.g. the first SIX multiples of:

**4** are 4, 8, 12, 16, 20, 24 …

**5** are 5, 10, 15, 29, 25, 30 …

**6** are 6, 12, 18, 24, 30, 36 …

**7** are 7, 14, 21, 28, 35, 42 …

From the examples above, we observed that 4, 5, 6, 7 are the first multiples of **4, 5, 6, 7 **respectively. This means that a number is also a multiple of itself.

**COMMON MULTIPLES**

When the multiples of two or more numbers are listed, the ones that are common are the common multiples.

** **

**Examples**

Give two common multiples of the following:

1. 2 and 3.

** Solution**

** list the multiples of 2 and 3

2 – 2, 4, **6**, 8, 10, **12**, 14, **18**

3 – 3, **6**, 9, **12**, 15, **18**, 21, 24

Common multiples are 6 and 12. 18 is also a common multiple, since we are looking for just two, we picked 6 and 12 being the first two.

** to get your common multiples easily, list the multiples of each number simultaneously i.e. as you write one multiple for 2, write one for 3 also.

2. 5 and 8

** Solution**

5 – 5, 10, 15, 20, 25, 30, 35, **40**, 45, 50, 55, 60, 65, 70, 75, **80**

8 – 8, 16, 24, 32, **40**, 48, 56, 64, 72, **80**

Common Multiples are 40 and 80

** Exercise G**

Give the first two common multiples of the following

1. 4 and 9

2. 3 and 7

3. 2, 3 and 5

4. 6 and 8

5. 4, 5 and 6

** LOWEST COMMON MULTIPLES (L.C.M)**

The LOWEST COMMON MULTIPLES written as L.C.M for short is the least of all the common multiples of the given numbers. It is also known as least common denominator. E.g. find the L.C.M of 9 and 15.

9 – 9, 18, 27, 36, **45**

15 – 15, 30, **45**

** once you see the first common multiple, stop immediately.

L.C.M is 45.

Using this method for getting lcm becomes very difficult when large numbers are involved. Using the index or table method just like that of H.C.F is highly recommended.

** Index method**

9 – 3^{2}

15 – 3 x 5

** in L.C.M, all bases must be selected unlike in H.C.F where you select only common bases.

** when a base is common to the two numbers, pick the one with the **highest** power. This is unlike in H.C.F where you pick the one with the smallest power.

From the index above, we have just two bases 3 and 5. The two will be selected. Base 3 has powers 2 and 1, we pick power 2 (being the higher)

Then L.C.M = 3^{2 }x 5

= 9 x 5

= 45

** TABLE METHOD**

9 | 15 |

Unlike the H.C.F, we look for numbers that can divide one of the given numbers. We continue to divide the two numbers (9 and 15) till they get to 1.

** we start with 2, 2 cannot divide any of them (9 and 15). Then we try 3

** 3 is a factor. It will give 3 and 5

3 | 9 | 15 |

3 | 5 |

** we try 3 again, it will divide 3 to give 1. Since it is not a factor of 5, we write the 5 again

3 | 9 | 15 |

3 | 3 | 5 |

1 | 5 |

** we now divide by 5

3 | 9 | 15 |

3 | 3 | 5 |

5 | 1 | 5 |

1 | 1 |

** LCM is the product of the numbers on the first column

LCM = 3 x 3 x 5

LCM = 45.

** Example 2**

Find the LCM of 24, 36 and 16.

** **

**Index method**

24 = 2^{3} x 3

36 = 2^{2} x 3^{2}

16 = 2^{4}

Here we have bases 2 and 3. Select the one with the highest power.

Lcm = 2^{4 }x 3^{2}

Lcm = 16 x 9

Lcm = 144

**Table method**

** **

** 2 can divide 24, 36 and 16 to give 12, 18 and 8 respectively

2 | 24 | 36 | 16 |

12 | 18 | 8 |

** 2 can still divide 12, 18 and 8 to give 6, 9 and 4 respectively

2 | 24 | 36 | 16 |

2 | 12 | 18 | 8 |

6 | 9 | 4 |

** 2 can still divide 6 and 4 while others remain

2 | 24 | 36 | 16 |

2 | 12 | 18 | 8 |

2 | 6 | 9 | 4 |

3 | 9 | 2 |

** 2 is not a factor of 3 and 9, but can still divide 2

2 | 24 | 36 | 16 |

2 | 12 | 18 | 8 |

2 | 6 | 9 | 4 |

2 | 3 | 9 | 2 |

3 | 9 | 1 |

** looking at the numbers 3 and 9, 2 cannot divide any without remainder, so we try 3.

2 | 24 | 36 | 16 |

2 | 12 | 18 | 8 |

2 | 6 | 9 | 4 |

2 | 3 | 9 | 2 |

3 | 3 | 9 | 1 |

1 | 3 | 1 |

** one more 3

2 | 24 | 36 | 16 |

2 | 12 | 18 | 8 |

2 | 6 | 9 | 4 |

2 | 3 | 9 | 2 |

3 | 3 | 9 | 1 |

3 | 1 | 3 | 1 |

1 | 1 | 1 |

**Once we get to 1, 1, 1 for the three numbers, we multiply the first column to get the lcm.

Lcm = 2 x 2 x 2 x 2 x 3 x 3

Lcm = 2^{4} x 3^{2}

Lcm = 16 x 9

Lcm = 144.

** Example 3**

Find the lcm of 35, 50 and 21

**Index method**

35 = 5^{1} x 7^{1}

50 = 2^{1} x 5^{2}

49 = 7^{2}

We have 3 bases here; 2, 5 and 7. Each base with the highest power will be selected.

Lcm = 2^{1 }x 5^{2} x 7^{2}

Lcm = 2 x 25 x 49

Lcm = 2450

**Table method**

** **

2 | 35 | 50 | 49 |

5 | 35 | 25 | 49 |

5 | 7 | 5 | 49 |

7 | 7 | 1 | 49 |

7 | 1 | 1 | 7 |

1 | 1 | 1 |

** **

Lcm = 2 x 5 x 5 x 7 x 7

Lcm = 2 x 5^{2} x 7^{2}

Lcm = 2 x 25 x 49

Lcm = 2450

** Exercise H**

Find the lcm of the following

1. 8 and 12

2. 6, 8 and 10

3. 15, 21 and 25

4. 48, 50 and 60

5. 14, 9 and 21

6. 30, 35 and 40

7. 29, 15 and 20

8. 12, 14, 16 and 18

9. 84 and 72

10. 145, 180 and 225

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