Real Numbers

Lowest Common Multiple

Do you know how scientists at NASA figure out when various planets will align with each other in our Solar system. Well, they use the basic principal of Lowest Common Multiple(LCM). Don’t believe us? Read ahead to find out…

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What is Lowest Common Multiple?

Let’s start with what a multiple is. So when you recite your timetables, all you’re doing is listing the multiples of that number. So the timetable of 3, i.e. 3,6,9,12,15,18….etc, are nothing but it’s multiples. And a Lowest Common Multiple(LCM) is nothing but the smallest whole number that is multiple of both the given numbers.

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Example:

Multiples of 3 are 3,6,9,12,15,18,21….

Multiples of 5 are 5,10,15,20,25,30….

LCM (3,5) = 15

Methods of calculating Lowest Common Multiple

Listing all the multiples

This is the simplest method of finding the LCM of any given numbers. You write down several multiples of both the numbers and then identify the smallest common multiple among them.

Example:

Multiples of 5 = 5,10,15,20,25,30.35,40,45,50….

Multiples of 8 = 8.16.24.32.40.48.56.64…..

LCM (5,8) = 40

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Prime factorization

We start by listing all the prime factors of a number. Then you take the common prime factors out, and then the prime factors which are not common. Multiply all these factors and you will arrive at the Least Common Multiple.

Example:

10 = 2*5

12 = 2*2*3

Uncommon Prime Factors = 2,3,5

LCM(10,12) = 2*2*3*5 = 60

Long Division Method

Lowest Common Multiple

Step 1 Write all the numbers in the first row divided by commas
Step 2 We divide the numbers by the lowest and most suitable prime number (should exactly divide at least 2 numbers)
Step 3 Write the quotient of the division in the next row, If the number is not exactly divisible by the prime number, bring it down as it is
Step 4 Continue to do the above steps, till only co-prime numbers are left in the last row.
Step 5 Multiply all prime numbers by which we have divided and all co-prime numbers left in the last row, This is your LCM

 

Finding Least Common Multiple of Decimals

We follow the same method of prime factorization, with a few changes

Example:

Let’s say we have to find the LCM of 2.5 and 0.35

First, we convert both numbers to like decimals i.e. 250 and 35

Now we express those two numbers as a product of their prime factors

250 = 2*5*5*5

35= 5*7

LCM (250,35) = 2*5*5*7*5 = 1750

Therefore LCM (2.5, 0.35) = 17.50

Solved Example for You

Question 1: What is the smallest number that when divided by 20 and 48 separately gives the remainder of 7 every time?

Answer : The solution here is an application of LCM principal. Here firstly we find the LCM of 20 and 48

20 = 2*2*5

48 = 2*2*2*2*3

LCM (20,48) = 2*2*2*2*3*5 = 240

So the required number, that leaves a remainder of 7, is 247 (240+7)

Question 2: How can one find the lowest common multiple?

Answer: One way to calculate the least common multiple of two numbers involves, first of all, listing the prime factors belonging to each number. Afterwards, one must multiply each factor the greatest number of times its occurrence takes place in either number. If the same factor appears multiple number of times in both numbers, you must multiply the factor the greatest number of times it occurrence takes place.

Question 3: What is meant by LCM?

Answer: The Least Common Multiple (LCM) refers to the smallest number that happens to be a multiple of all the numbers. For instance, the LCM of the numbers 16 and 20 shall be 80.

Question 4: What will be the LCM of the numbers 9, 12, and 15?

Answer: The LCM of the numbers 9, 12, 15 is the consequence of multiplying the greatest number of times all prime factors that occur in these numbers. The LCM of the numbers 9, 12, and 15 will be 2⋅2⋅3⋅3⋅5=180 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 = 180.

Question 5: Find the LCM of numbers 4 and 5?

Answer: The LCM of the numbers 4 and 5 shall be 20.

 

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