How to find the LCM of 4 numbers?

Before going to find Least Common Multiple we have to know the number system. In this number system, we can learn natural numbers, whole numbers, Integers, and Rational numbers. If we have an idea about the number system we can be able to identify the type of number it is? so that we can find LCM easily and we will get an idea of what numbers we have to find LCM and what is the easiest way to find factors or divisors to it.

For example, take numbers that start from 1 to infinite they are natural numbers, numbers include zero are whole numbers, numbers start with having negatives and positives, and also zero are integers, numbers in the form of a/b are rational numbers.

What is the Number System?

A Number system is a way of representing or expressing in writing the numbers like using numbers in a mathematical notation. The number is like counting the objects or digits in our real-world activities.

Example: Natural numbers set is represented as {1, 2, …} are numbers which we are called natural numbers. Like natural numbers, we have Whole numbers, Integers, Complex numbers, and Rational numbers.

• Natural Numbers: Natural numbers are counting numbers like 1,2,3,4 and so on which are positive numbers only and 0 is not a natural number.
• Whole numbers: Whole numbers are having zero included with natural numbers. A set of 0,1,2,3 and so on are whole numbers that are positive only no negative numbers are included in these whole numbers. All Natural numbers can be whole numbers but all whole numbers need not be natural numbers.
• Integers: Integers are having negative numbers as well as whole numbers. A set of {…, -2, -1, 0, 1, 2, 3, …} are Integers. All whole numbers are Integers but all integers are not whole numbers.
• Rational numbers: Rational numbers include all integers, fractions, and decimals. A rational number can be represented as a/b form. 3 can be represented as 3/1. So it is a rational number and also 3.33 can be represented as 3.33/1.

What is LCM? How to find it?

LCM is abbreviated as a Least Common Multiple. The Least Common Multiple of four numbers or any number of numbers is the smallest common number which is non zero and also it should be a multiple of numbers for which we are finding the Least Common Multiple. Multiple is when we are dividing a given number with a number and leaving no remainder is called multiple of the given number.

For Example, Suppose we have a number 16 which can be divided by 1, 2, 4, 8, and 16 itself are various multiples for a given number 16.

• 16 = 1×16 or (16/1=16 and also 16/16 = 1 so that there is no remainder while dividing are called multiples. 1,16 are multiples)
• 16 = 2×8 or (16/2=8 and also 16/8 = 2 where 2 and 8 are multiples of 16)
• 16 = 4×4 or (16/4 = 4 where 4 is a quotient as well as multiple but no remainder while dividing)

Methods to determine the LCM

There are different methods to find LCM of two or more numbers:

1. LCM using Prime Factorization Method.
2. LCM using repeated division or Long division.
3. LCM using multiples of a number. 

LCM using Prime Factorization Method

Before going into this we have to know factorization and that to prime factorization. Prime numbers are the numbers that have only two factors are 1 and themselves.

For Example, 3 is a prime number because 3 can have factors 1 and 3. 

• 1 is a factor of 3 because 1×3 is 3
• 3 is a factor of 3 because 3×1 is 3

Prime Factorization is a way of making numbers a product of prime numbers.

LCM of 10,12,14,16 using prime factorization is expressed as,

LCM using prime factorization

• 10 = 2 × 5
• 12 = 2 × 2 × 3
• 14 = 2 × 7
• 16 = 2 × 2 × 2 × 2

In the above factors of each and every number, only 2 is common in all numbers so 2×2×2×2×3×5×7 = 1680 is the LCM of 10, 12, 14, 16.

LCM using Repeated division or Long division Method.

In this method, numbers are divided with the common divisors until no further possible division occurs. Divisors are the numbers in which we have to divide a number with another number that another number is called the divisor.

For Example, Suppose the number 12 is divided by either 1, 2, 3, 4, 6, or 12.

• 12 = 2×6 because we can able to written 12 as 2×6 so 2 and 6 are divisors of 12.
• 12 = 4×3 because we can able to write 12 as 4×3 so 4 and 3 are divisors of 12.
• 12 = 12×1 because of this we can able to write as 12×1 so, 1, 12 are divisors of 12.

LCM of 10,12,14,16 using Repeated division is expressed below.

LCM using repeated division

So from here if we multiply divisor and remainder we get LCM of 10, 12, 14, 16.

LCM = 2×2×2×5×3×7×2 = 1680

LCM using multiples of a number

Finding LCM using multiples is the selecting first most common multiple among the group of multiples of numbers. When we are dividing a given number with a number and leaving no remainder is called a multiple of the given number.

For Example, LCM of 2,4,6,8 is the first common multiple among the multiples of 2, 4, 6, 8.

LCM using Multiples 

In the above figure, it is the table form of multiples of 2, 4, 6, 8. In the above table, 24 is the LCM of 2, 4, 6, 8 because it is the first common multiple of all the numbers 2, 4, 6, 8.

Properties of LCM

• The L.C.M. of at least two numbers can’t be not exactly or less than any of them. For example, L.C.M of 3,4,5,6 is 60 which is not less than any of the given numbers(3,4,5,6).
• The factor of a number and their L.C.M is greater than the number itself. For example, L.C.M of 4,8 is 8.

Sample Questions 

Question 1: What is the LCM of the series of numbers 10, 20, 30, and 40?

Answer:

This question is solved by using the repeated division method as,

10, 20, 30, 40 can be written as follows:

• 10 = 2×5 where 2 and 5 are prime numbers.
• 20 = 2×10=2×2×5, so here also 10 again written as 2×5 so all are prime numbers. 

Similarly, 

• 30 can be written as 30=3×10 and again 10 can be written as 2×5 so 2, 3, and 5 are prime numbers.
• 40 = 2×20=2×2×10=2×2×2×5.

LCM in a detailed way for this problem is shown as:

LCM(10,20,30,40)=120

The above method of finding LCM is Repeated division discussed in the types for finding LCM.

Therefore, LCM (10, 20, 30, 40) = 120 (Since, 2×2×5×2×3×1×1 is 120).

Question 2: What is the LCM of the series of numbers 2, 3, 5, and 7?

Answer:

Here 2, 3, 5, and 7 all are prime numbers so LCM is simply a product of all the given numbers that is 2×3×5×7 is the LCM.

By using repeated division 2 can be written as 2×1, similarly 3=3×1,5=5×1,7=7×1 like that.

So. LCM of 2, 3, 5, 7 is 2×3×5×7=210

Detailed LCM of this problem is given as.

The above method of finding LCM is Repeated division discussed in the types for finding LCM.

Therefore LCM (2, 3, 5, 7) = 210 (Since 2×5×7×3×1 is 210).

Question 3: What is the LCM of the series of numbers 1, 2, 3, and 4?

Answer:

Let’s try the multiples method for this question. For this, we have to write multiples of each and every number means we have to write a table of 1, 2, 3, 4 until the first most common multiple occurs.

• 1 multiples are 1×1=1, 1×2=2, 1×3=3, and so on.
• 2 multiples are 2×1=2, 2×2=4, 2×3=6, 2×4=8, and so on.
• 3 multiples are 3×1=3, 3×2=6, 3×3=9, 3×4=12, and so on.
• 4 multiples are 4×1=4, 4×2=8, 4×3=12, and so on.

Detailed LCM of this problem is given as,

We have to write multiples in that table until the most common factor in all the numbers occurs.

LCM (1,2,3,4) =12

The above method of finding LCM is using multiples discussed in the types for finding LCM.

Therefore, LCM (1, 2, 3, 4)=12 (because the least common multiple among all multiples is 12).

Question 4: What is the LCM of the series of numbers 99, 66, 33, and 11?

Answer:

Prime factorization contains a prime number we have to write for a series of numbers.

• 99 can be written as 11×9 and again 9 can be written as 3×3 so finally 99=11×3×3 where 11,3,3 are prime numbers.
• 88 can be written as 11×6 and again 6 can be written as 2×3 so finally, 66 =11×2×3 where 11,2,3 are prime numbers.
• 33 can be finally written as 11×3 where 11,3 are prime numbers.
• 11 itself can be written as 11×1 where 11 is a prime number.

Detailed LCM of this problem is given as,

LCM (11,33,66,99) =198

The above method of finding LCM is the prime factorization discussed in the types for finding LCM.

Therefore, LCM(99, 66, 33,11) = 198 (because 11×3×2×3 is 198).

Question 5: What is the LCM of the series of numbers 0, 25, 16, 36?

Solution:

Prime factorization contains a prime number we have to write for a series of numbers.

• 4 can be written as 2×2 where 2 is a prime number.
• 6 can be written as 2×3 where 2,3 are prime numbers.
• 25 can be finally written as 5×5 where 5 is a prime number.
• 0 itself can be written as 0×1 

Detailed LCM of this problem is given in the below image.

LCM(0,4,6,25)=300

The above method of finding LCM is the prime factorization discussed in the types for finding LCM.

Therefore LCM(4, 6, 25, 0)=300 (because 2×2×5×5×3 is 300).