LEAST COMMON MULTIPLE BY DIVISION METHOD

Least common multiple is the smallest number which is a multiple of the given numbers.

For example, consider the numbers 4 and 5.

The whole numbers which are multiples of both both 4 and 5 are

20, 40, 60, 80, ..........

All the above numbers are multiples of both 4 and 5.

Can we get a whole number less than 20 which is a multiple of both 4 and 5?

NO, we CAN'T.

So, 20 is the smallest whole number which is a multiple of both 4 and 5.

In other words, 20 is the smallest whole number which is evenly divisible by both 4 and 5.

Therefore, 20 is the least common multiple of 4 and 5.

More Examples :

Least common multiple of 2 and 3 = 6

Least common multiple of 4 and 6 = 12

Least common multiple of 3 and 6 = 6

Least Common Multiple of Two Number by Division Method

Using L division, divide both the numbers by a common divisor.

Continue this process, until you get a common divisor for both the numbers.

At one stage, there will be no common divisor for both the numbers.

Then, multiply all the common divisors and the numbers at the last stage to get the least common multiple of the given two numbers.

Note :

Usually prime numbers are being used as divisors.

Example 1 :

Find the least common multiple of 24 and 60.

Solution :

Using L division, divide both the numbers by a common divisor. Continue this process, until you get a common divisor for both the numbers.

lcm3.png

The least common multiple of 24 and 60 is

= 2 x 2 x 3 x 2 x 5

= 120

Least Common Multiple of More Than Two Numbers by Division Method

Consider the least common multiple of three numbers.

Try to get a common divisor for all the given three numbers.

If you get a common divisor for all the three numbers, using L division, divide all the the three numbers by it.

If you don't get a common divisor for all the three numbers, try to get a common divisor for at least two numbers and divide those two numbers by it. Write the third number as it is.

Continue this process, until you get a common divisor for at least two numbers.

At one stage, there will be no common divisor even for two numbers.

Then, multiply all the common divisors and the numbers at the last stage to get the least common multiple of the given three numbers.

You can follow the same process to find the least common multiple for than three numbers.

Example 2 :

Find the least common multiple of 48, 72 and 108.

Solution :

Using L division, divide the given numbers by a common divisor. Continue this process, until you get a common divisor at least for any of the two numbers.

lcm4.png

The least common multiple of 48, 72 and 108 is

= 2 x 2 x 3 x 2 x 3 x 2 x 1 x 3

= 432

Solved Problems

In each case, find the least common multiple of the numbers by division method.

Problem 1 :

4 and 6

Solution :

Using L division, divide both the numbers by a common divisor. Continue this process, until you get a common divisor for both the numbers.

lcm5.png

The least common multiple of 4 and 6 is

= 2 x 2 x 3

= 12

Problem 2 :

2 and 3

Solution :

There is no common divisor for both 2 and 3 other than 1.

So, the least common multiple of 2 and 3 is

= 2 x 3

= 6

Problem 3 :

4 and 5

Solution :

There is no common divisor for both 4 and 5 other than 1.

So, the least common multiple of 4 and 5 is

= 4 x 5

= 20

Problem 4 :

3 and 6

Solution :

Using L division, divide both the numbers by a common divisor. Continue this process, until you get a common divisor for both the numbers.

lcm6.png

The least common multiple of 4 and 6 is

= 3 x 1 x 2

= 6

Problem 5 :

12 and 18

Solution :

Using L division, divide both the numbers by a common divisor. Continue this process, until you get a common divisor for both the numbers.

lcm7.png

The least common multiple of 12 and 18 is

= 2 x 3 x 2 x 3

= 36

Problem 6 :

4, 5 and 8

Solution :

Using L division, divide the given numbers by a common divisor. Continue this process, until you get a common divisor at least for any of the two numbers.

lcm8.png

The least common multiple of 4, 5 and 8 is

= 2 x 2 x 1 x 5 x 2

= 40

Problem 7 :

4, 6, 8 and 12

Solution :

Using L division, divide the given numbers by a common divisor. Continue this process, until you get a common divisor at least for any of the two numbers.

lcm9.png

The least common multiple of 4, 6, 8 and 12 is

= 2 x 2 x 3 x 1 x 1 x 2 x 1

= 24

Problem 8 :

2, 3, 4 and 5

Solution :

Using L division, divide the given numbers by a common divisor. Continue this process, until you get a common divisor at least for any of the two numbers.

lcm10.png

The least common multiple of 2, 3, 4 and 5 is

= 2 x 1 x 3 x 2 x 5

= 60

Problem 9 :

2, 4, 5 and 8

Solution :

Using L division, divide the given numbers by a common divisor. Continue this process, until you get a common divisor at least for any of the two numbers.

lcm11.png

The least common multiple of 2, 4, 5 and 8 is

= 2 x 2 x 1 x 1 x 5 x 2

= 40

Problem 10 :

3, 6, 7 and 14

Solution :

Using L division, divide the given numbers by a common divisor. Continue this process, until you get a common divisor at least for any of the two numbers.

lcm12.png

The least common multiple of 3, 6, 7 and 14 is

= 3 x 7 x 2 x 1 x 1 x 1 x 1

= 42

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