Introduction to Least common number or least common multiple

In arithmetic and number theory, the least common number or least common multiple or lowest common multiple (LCM) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both of a and of b. Since it is a multiple, it can be divided by a and b without a remainder.

The definition is sometimes generalized for more than two integers: The lowest common multiple of integers a1, ..., an is the smallest positive integer that is a multiple of a1, ..., an

 

Example problems

 

Example 1

Find the least common number of 4, 11, and 44

Solution

To find the least common number of 4, 11, and 44, first list out the multiples of 4, 11, and 44 individually.

Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52

Multiples of 11 = 11, 22, 33, 44, 55, 66

Multiples of 44 = 44, 88, 132

Here, 44 is the common number in the multiples of 4, 11, and 44.

So, 44 is the least common number or multiple of 4, 11, and 44.

Example 2

Find the least common number of 25 and 35.

Solution

To find the least common number of 25 and 35, first list out the multiples of 25 and 35 individually.

Multiples of 25 = 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275

Multiples of 35 = 35, 70, 105, 140, 175, 210, 245, 280

Here 175 is the common number in the list of multiples of 25 and 35.

So, 175 is the least common number or least common multiple of 25 and 35.

 

 

Least common denominator

 

Least common number is very useful in the addition or subtraction of fractions as least common denominator. We know that, we can only add fractions, whose denominators are same. If we have to add two fractions with different denominators, least common denominator is used to convert the fractions into like fractions(same denominators).

Example problem

`1/5` + `2/7` = ?

Solution

Here, the denominators of the fractions are different (5 and 7).

To make these unlike fractions in to like fractions, we need a common denominator.

The least common denominator of 5 and 7 is 35.

So, the fractions,

`1/5` becomes `7/35`

and `2/7` becomes `10/35`

Now, `7/35` and `10/35` are like fractions.

So, `7/35` + `10/35` = `(7 + 10)/35`

                       = `17/35`

So, `1/5` + `2/7` = `17/35`