Lowest Common Multiple (LCM)

What is LCM?

The lowest common multiple of two or more numbers is the lowest multiple they have in common. 

For example:

The multiples of 2 are  2, 4, 6, 8, 190, 12, 14, 16, 18……

The multiples of 3 are 3, 6, 9, 12, 15, 18 …….

The common multiples of 2 and 3 are 6, 12, 18, ……

The lowest common multiple of 2 and 3 is 6.

Method 1 (Short Division Method):

LCM by division method means finding the least common multiple of a given set of numbers by dividing all the given numbers by a common prime number.

Method 2 (Prime Factorization Method):

Using this method we find the prime factors of the given numbers. We the select each prime factor with the “highest power” and multiply to get the LCM.

Let’s try some examples.

Worked Example 1.4.1:

Find the LCM of:

a. 3, 7 and 10

b. 9 and 15

c. 2 ,3 and 4

d.
2³ × 3² × 5
2² × 3 × 5

e.
2⁴ × 3² × 5 × 7 × 11²   
2³ × 5 × 7³ × 11
2² × 3³ × 5 × 7² × 11

a. 3, 7 and 10

Solution

1st Method (Division Method):

Let us understand this method using the example given below.

Step 1: 2 is the smallest prime number and it is a factor of 10. Write 2 on the left of the three numbers. For each number in the right column, continue calculating out prime numbers which are their factors.

Step 2: 2 divides 10 and the result is 5. But 2 is not a factor of 3 and 7, so we write the number 3 and 7 in the row below as it is.

Step 3: We continue the process until we have only 1’s left

Step 4: Our final step is to multiply the prime numbers on the left. The Least Common Multiple (LCM) is the product of all these prime numbers.

LCM  = 2 × 3 × 5 × 7

= 6 × 35

=  210

2nd Method:

Find the prime factors of the given numbers.

3 = 1 × 3
7 = 1 × 7
10 = 2 × 5

Select each prime factor with the “highest power” and multiply to get the LCM.

∴ L.C.M. = 1 × 2 × 3 × 7 × 5

= 210

b. 9 and 15

Solution

1st Method (Division Method):

∴ L.C.M. = 3 × 3 × 5

= 9 × 5

= 45

2nd Method (Prime Factorization Method):

Step 1: Resolve each given number into its prime factors and express the factors obtained in exponent form.

9 = 3 × 3 = 3²

15 = 3 × 5

Step 2: Find the product of all the prime factors with highest powers.

= 3² × 5

= 9 × 5

= 45

c. 2, 3 and 4

Solution

LCM of 3 and 4 = 2 × 2 × 3 = 12

d.

2³ × 3² × 5
2² × 3 × 5

Solution

In this case, compare the index (power) of numbers with the same base. The numbers with greater power (index) are the common multiple.

2³ × 3² × 5

2² × 3 × 5

= 2³  ×   3²   ×   5

LCM   =   2³   ×  3²   ×  5

e.

2⁴ × 3² × 5 × 7 × 11² 
2³ × 5 × 7³ × 11
2² × 3³ × 5 × 7² × 11²

Solution

LCM   =   2⁴   ×  3³   ×  5  ×  7³   ×  11²

Evaluation Questions: