We will discuss here about the method of l.c.m. (least common multiple).

**1. Let us consider the numbers 8, 12 and 16.**

Multiples of 8 are → 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, ......

Multiples of 12 are → 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, ......

Multiples of 16 are → 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, ......

The common multiple of 8, 12, 16 are 78, 96, ......

The least common multiple of 8, 12 and 16 is 48. (Smallest common multiple)

In short, the lowest common factor is expressed as L.C.M.

**2. Find the L.C.M. of 3 and 4.**

Multiples of 3 = 3, 6, 9, __12__, 15, 18, 21, __24__, ............

Multiples of 4 = 4, 8, __12__, 16, 20, __24__, 28, ............

Common multiples of 3 and 4 = 12, 24, ............

Least common multiple of 3 and 4 = 12.

**3. Find the L.C.M. of 6 and 12.**

Multiples of 6 = 6, __12__, 18, __24__, 30, __36__, 42, __48__, ............

Multiples of 12 = __12__, __24__, __36__, __48__, 60, 72, 84, ............

Common multiples of 6 and 12 = 12, 24, 36, 48, ............

Least common multiple of 6 and 12 = 12.

** **

Finding L.C.M.

**Least Common Multiple (L.C.M.) by Prime Factorisation Method:**

To find the L.C.M. we find prime factors of the given numbers.

Remember, we consider common prime factors only.

**Solved Examples:**

**1. Find the L.C.M. of 12, 16 and 24.**

First we find the prime factors of the given numbers.

12 = 2 × 2 × 3

16 = 2 × 2 × 2 × 2

24 = 2 × 2 × 2 × 2 × 3

(2 comes maximum 4 times and 3 comes maximum once only.)

L.C.M. = 2 × 2 × 2 × 2 × 3

= 48 which is the product of their prime factors.

**2. Find the L.C.M. of 10 and 16.**

10 = 2 × 5 16 = 2 × 2 × 2 × 2 Common factor = 2 Other factors = 2, 2, 2, 5 |
2 | 5 |
2 | 2 | 2 | 2 |

Therefore, L.C.M. = 2 × 2 × 2 × 2 × 5 (common factor × other factors)

= 80

20 = 2 × 2 × 5 25 = 5 × 5 Common factor = 5 Other factors = 2, 2, 5 |
2 | 2 | 5 |
5 | 5 |

Therefore, L.C.M. = 5 × 2 × 2 × 5 (common factor × other factors)

= 100

**4. Find the L.C.M. of 16 and 24.**

16 = 2 × 2 × 2 × 2 24 = 2 × 2 × 2 × 3 Common factor = 2, 2, 2 Other factors = 2, 3 |
2 | 2 | 2 | 2 |
2 | 2 | 2 | 3 |

Therefore, L.C.M. = 2 × 2 × 2 × 2 × 3 (common factor × other factors)

= 48

**Least Common Multiple (L.C.M.) by Division Method:**

We can also find the L.C.M. of the given numbers by dividing all the numbers at the same time by a number that divides at least two of the given numbers.

1. When a number is not exactly divisible, we write the number itself below the line. 2. When we cannot divide the numbers by a common factor exactly we discontinue dividing the numbers. |

L.C.M. = 2 × 2 × 2 × 3 × 2 = 48

**Note: **

The product of L.C.M. and H.C.F. of two numbers is also the product of the numbers.

For example, the L.C.M. of 7 and 14 is 14 and the H.C.F. of 7 and 14 = 7. We see that the product of 7 and 14 also the product of L.C.M. and H.C.F. of 7 and 14.

**1. Find the L.C.M. of 25 and 45.**

Therefore, L.C.M. = 5 × 5 × 9

= 225

**2. Find the L.C.M. of 40, 68 and 72.**

Divide 40, 68 and 72 by 2. 40 ÷ 2 = 20; 68 ÷ 2 = 34; 72 ÷ 2 = 36 Divide 20, 34 and 36 by 2. 20 ÷ 2 = 10; 34 ÷ 2 = 17; 36 ÷ 2 = 18 10, 17 and 18 do not have a common factor. But 10 and 18 have 2 as a common factor. Divide 10 and 18 by 2, leaving 17 as it is. 10 ÷ 2 = 5; 18 ÷ 2 = 9 5, 17 and 9 do not have a common factor. Stop the division. |

Therefore, L.C.M. = 2 × 2 × 2 × 5 × 17 × 9

= 6120

**Questions and Answers on Method of LCM:**

**I. Find the L.C.M. of the following by prime factorisation method.**

(i) 30, 36

(ii) 12, 15

(iii) 5, 7

(iv) 15, 30

(v) 42, 72

(vi) 12, 48

(vii) 60, 75

(viii) 25, 150

(ix) 64, 128

(x) 60, 108

**Answer:**

**I. **(i) 180

(ii) 60

(iii) 35

(iv) 30

(v) 504

(vi) 48

(vii) 300

(viii) 150

(ix) 128

(x) 540

**II. Find the L.C.M. of the following by division method.**

(i) 27, 84

(ii) 16, 32

(iii) 12, 15

(iv) 25, 30

(v) 60, 70

(vi) 30, 18, 60

(vii) 88, 64, 96

(viii) 48, 96, 144

(ix) 26, 28, 24

(x) 16, 12, 20

**Answer:**

**II. **(i) 756

(ii) 32

(iii) 60

(iv) 150

(v) 420

(vi) 180

(vii) 2112

(viii) 288

(ix) 2184

(x) 240

**III. Find the L.C.M. of the following numbers by listing the multiples.**

(i) 24, 36

(ii) 12, 18

(iii) 10, 20, 40

(iv) 27, 108

(v) 63, 84

**Answer:**

**III. **(i) 72

(ii) 36

(iii) 40

(iv) 108

(v) 252

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