Least Common Multiple

The least common multiple (L.C.M.) of two or more numbers is the smallest number which can be exactly divided by each of the given number.

Let us find the L.C.M. of 2, 3 and 4.

Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, ...... etc.

Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ...... etc.

Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, ...... etc.

Common multiples of 2, 3 and 4 are 12, 24, 36, ...... etc.

Therefore, the smallest common multiple or least common multiples of 2, 3 and 4 is 12.

We know that the lowest common multiple or LCM of two or more numbers is the smallest of all common multiples.

Let us consider the numbers 28 and 12

Multiples of 28 are 28, 56, 84, 112, …….

Multiples of 12 are 12, 24, 36, 48, 60, 72, 84, …….

The lowest common multiple (LCM) of 28 and 12 is 84.

Let us consider the first six multiples of 4 and 6.

The first six multiples of 4 are 4, 8, 12, 16, 20, 24

The first six multiples of 6 are 6, 12, 18, 24, 30, 36

The numbers 12 and 24 are the first two common multiples of 4 and 6. In the above example the least common multiple of 4 and 6 is 12.

Hence, the least common multiple or LCM is the smallest common multiple of the given numbers.

Consider the following.

(i) 12 is the least common multiple (L.C.M) of 3 and 4.

(ii) 6 is the least common multiple (L.C.M) of 2, 3 and 6.

(iii) 10 is the least common multiple (L.C.M) of 2 and 5.

We can also find the L.C.M. of given numbers by their complete factorisation.

To find for instance, L.C.M. of 24, 36 and 40, we first factorise them completely.

24 = 2 × 2 × 2 × 3 = 2$^{3}$ × 3$^{1}$

36 = 2 × 2 × 3 × 3 = 2$^{2}$ × 3$^{2}$

40 = 2 × 2 × 2 × 5 = 2$^{3}$ × 5$^{1}$

L.C.M. is the product of highest power of primes present in the factors.

Therefore, L.C.M. of 24, 36 and 40 = 2$^{3}$ × 3$^{2}$ × 5$^{1}$ = 8 × 9 × 5 = 360

$Lowest Common Multiple$

Solved examples to find the lowest common multiple or the least common multiple:

1. Find the L.C.M. of 8, 12, 16, 24 and 36

8 = 2 × 2 × 2 = 2$^{3}$

12 = 2 × 2 × 3 = 2$^{2}$ × 3$^{1}$

16 = 2 × 2 × 2 × 2 = 2$^{4}$

24 = 2 × 2 × 2 × 3 = 2$^{3}$ × 3$^{1}$

36 = 2 × 2 × 3 × 3 = 2$^{2}$ × 3$^{2}$

Therefore, L.C.M. of 8, 12, 16, 24 and 36 = 2$^{4}$ × 3$^{2}$ = 144.

2. Find the LCM of 3, 4 and 6 by listing the multiples.

Solution:

The multiple of 3 are 3, 6, 12, 15, 18, 21, 24

The multiple of 4 are 4, 8, 12, 16, 20, 24, 28

The multiple of 6 are 6, 12, 18, 24, 30, 36, 42

The common multiples of 3, 4 and 6 are 12 and 24

So, the least common multiple of 3, 4 and 6 is 12.

We can find LCM of given numbers by listing multiples or by long division method.

2. Find the LCM of 18, 36 and 72 by division method.

Solution:

Write the numbers in a row separated by commas. Divide the numbers by a common prime number. We stop dividing after reaching the prime number. Find the product of divisors and the remainders.

$LCM by Division Method$

So, LCM of 18, 36 and 72 is 2 × 3 × 3 × 1 × 2 × 4 = 432

Questions and Answers on Least Common Multiple:

I. Find the LCM of the given numbers. First one is shown for you as an example.

(i) 3 and 6

3 = 3, 6, 9, 12, 15, 18, 21, 24, 27 ………….

6 = 6, 12, 18, 24, 30, 36, 42 ………….

The common multiples of 3 and 6 are 6, 12, 18 ………….

Lowest common multiple of 3 and 6 is 6.

(ii) 2 and 4

(ii) 4 and 5

(iii) 3 and 12

(iv) 15 and 20

Answers:

I. (ii) 4

(ii20

(iii) 12

(iv) 60

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