# LCM

### What is Multiple?

If the remainder, when a number X is divided by another number x is zero, then X is said to be a multiple of x.

And the term LCM is the smallest common multiple of any two or given positive integer.

### LCM Definition

LCM or least common multiple is the simplest method to find out the smallest common multiples between two or more than two numbers. Generally, the common multiple is a number which is a multiple of two or more than two numbers. LCM is used to find out the least common factor or multiple of any two or more given integers. For example, LCM of 10 and 30 is 30, where 30 is the smallest common multiple for numbers 10 and 30.

Now if we consider the multiple of 10 (10, 20, 30, 40,50…)and multiples of 30 (30, 60, 80, 120), we can see the first common or smallest common multiple between both the numbers is 30.

### Highest Common Factor (HCF)

The highest common factor of two numbers is the largest number that exactly divides the two numbers. It is also called the greatest common divisor (GCD). For example, the greatest common divisor of 20 and 15 is 5, as the number 5 can exactly divide both the numbers 20 and 15.

### LCM Formula

Here, You Can See LCM Formula Which Will Help You to Calculate LCM of Two Integers:

Let x and y are two integers, we can write the LCM formula with respect to the greatest common divisor(gcd) as given below:

L.C.M. (x,y) = (x * y) / gcd (x,y)

This is the formula for finding the LCM of any two integers

But for a fraction, LCM formula becomes

LCM – L.C.M. of Numerator/ H.C.F. of Denominator

### LCM Example:

Find the Least Common Multiple of Two Integers 10 and 20

Solution: We know that for any two integers x and y,

LCM(x,y) =(x *y)/gcd(x,y)

Hence, LCM (10,20) = (10 *20)/gcd(10,20)

The greatest common divisor for 10 and 20 is 10

Thus , LCM (10,20) = 200/gcd(10)

L.C.M. (10,20) = 20

### Solved LCM Examples

1. Find the LCM of 1.05 and 2.1

The first step is to convert the decimal numbers into like decimals. Therefore, the numbers are 105 and 210.

Now, Find the LCM of 105 and 210

Prime factors of 105 are – 3 *5 *7

Prime factors of 210 are – 2 *3 *5 *7

Prime factors of 105 and 120 are -2 * 3* 5 *7* 3 *5* 7

We can  there are one pair of common factors of a number,3,5 and 7 and one uncommon factor i.e. 2

Now after  pairing the common factors and uncommon factor  – 2* 3* 5 *7= 210

We get,

LCM of 105 and 210 and  i.e. 210

In decimal form, LCM = 2.1 ( Taking 2 decimal places)

2. How often will 5 bells ring together in one hour if they start together and ring at the intervals of 5,6,8,12,20 seconds respectively?

Solution. The first step is to take out the LCM of 5,6,8,12 and 20 seconds to find the ttime at which all the 5 bells will ring together

LCM of 5,6,8,12 and 20 is 120

The number of times they will ring together in one hour = 3600/120 = 30 ( 1 hour = 3600 sec)

Hence, the bell will ring together 30 times in an hour.

3. Find the LCM of 2/9 and 8/21

Solution: LCM of the numerators 2 ,3 and 6 is 6

HCF of the denominators 5, 10 and 25 is 5

LCM = LCM of the Numerators / HCF of the Denominators

Hence, LCM of  2/3, 3/10 and 6/25 is 6/5

Facts

•  The wax table was used by the Greeks to record the multiplication table in the first century AD.

• The symbol “?” is first used by Wiliam Oughtred for multiplication in the 15th century to teach Maths.

### Quiz Time

1. Find the LCM of 2/3, 8/9, 64/81, 10/27

1. 250/9
2. 160/3
3. 128/9
4. 320/3

2. Find the LCM of 87 and 145

1. 1305
2. 435
3. 875
4. 48

3. Find the Least Number Which is Exactly Divisible by 12, 15, and 20.

1. 40
2. 50
3. 60
4. 80

1. What is Known as Multiples and Common Multiples?

Ans: A multiple is a result that is received by multiplying any number by an integer ( it should not be a function). We get the multiples of a whole number by taking out the product of any of the counting numbers and that of whole numbers.

For example, we can find out the multiples of 5 by multiplying the number 5 by 1, 5 by 2, 5 by 3, etc…

Common Multiples

The multiples that are common to two or more than two numbers are known as common multiples of those numbers.

For Example

Consider any two number 50 and 60

Multiples of 25 are = 25,50,75,100,125,150,175,200….

Multiples of 50 are – 50,100, 150, 200,250,300,50,400…

In the above example, we can see that 50,100, 150 and 200 are the first four common multiples of 25 and 50.

2. How to Find the LCM?

Ans: Two Methods to Find the LCM of Integers are Stated below:

1. By Finding the Multiples

2. By Prime Factorization.

By Finding the Multiples

In this method, the least common multiples can be found by writing down the multiples of individual numbers and then find the first common multiples between them. For example, there are two numbers 10 and 30. Then the multiples of 10 and 30 can be written as

Multiples of two digit number 10 = 10, 20, 30, 40, 50…

Multiples of two digit number 30 are -30 , 60, 90 ,120, 150 ….

The first common multiple or the least common multiple of both the numbers is 30. Hence the LCM of 10 and 30 is 30.

By Prime Factorization Method

The other method to find the least common multiple is the prime factorization method.

Let us take three numbers 12,16 and 24. The first step is to write the prime factors of all these three numbers individually.

12= 2*2*3

16= 2*2*2*2

24= 2*2*2*3

Now write the prime factors of all the three numbers togetherly,

12*16*24= 2*2*3* 2*2*2*2*2*2*2*3

Now, we will get the prime numbers by pairing the common prime factors,

Here, we can see there are four pairs of 2 and one pair of 3

So the LCM of 12 , 16 and 24 will be: 2*2*2*2*3 = 48