Lowest common multiple of 3 numbers: Least common multiple of three numbers (video)
Posted onHow to Find the LCM for 3+ Numbers — FAST!
Is there a quick-and-easy way to find the LCM for three or more numbers … WITHOUT prime factorizing?
Of course! We’ll demonstrate the technique by finding the LCM for 10, 14, 20.
High-Octane Boost for Math Ed
To begin, use the technique for finding the GCF for 10, 14, 20 that’s shown in my post: How to Find GCF for 3+ Numbers — FAST … no prime factorizing. If you don’t want to go to that post, no worries. I’ll re-show the technique here.
1st) Write the numbers from left to right:
………. 10 14 20
[The periods: …… are just to indent the lines. They have no mathematical meaning.]
2nd) If possible, rip out a factor common to all numbers. The factor 2 is common. So divide the three numbers by 2 [10 ÷ 2 = 5 and 20 ÷ 2 = 10] and show the result below:
2 | 10 14 20
………. 5 7 10
3rd) At this point, notice there’s no number that goes into the remaining numbers: 5, 7, 10. That means you’ve found that the GCF is the number pulled out, 2. At this point we’re at a crossroads. We’re done finding the GCF, but now we’re at the start of a new process, finding the LCM.
To proceed toward getting the LCM, see if there’s any number that goes into any pair of remaining numbers. Well, 5 goes into 5 and 10. So divide both those numbers by 5 [5 ÷ 5 = 1 and 10 ÷ 5 = 2] , and show the results below:
2 | 10 14 20
5 | 5 7 10
……….. 1 7 2
Notice that if there’s a number 5 doesn’t go into, you leave that number as is. So leave the 7 as 7.
4th) Repeat. See if there’s a number that goes into two of the remaining numbers. Since nothing goes into 1, 7, and 2, we’re done. To get the LCM, multiply all of the outer numbers. That means you multiply the numbers you pulled out on the left (2 and 5), and also multiply the numbers at the bottom (1, 7 and 2).
Ignoring the meaningless 1, you have: 2 x 5 x 7 x 2 = 140, and that’s the LCM.
To see the process in more depth, let’s find the LCM for … not three, not four … but five numbers:
6, 12, 18, 30, 36.
1st) Write the numbers left to right:
……… 6 12 18 30 36
2nd) If possible, rip out a common factor. 2 is common, so divide all by 2 and show the results below:
2 | 6 12 18 30 36
………. 3 6 9 15 18
3rd) Repeat. See if there’s a number that goes into the five remaining numbers. 3 goes into all, so divide all by 3 and show the results below:
2 | 6 12 18 30 36
3 | 3 6 9 15 18
…….. 1 2 3 5 6
4th) Repeat. See if any number goes into the last remaining numbers. Nothing goes into all of them, so now you get the GCF by multiplying the left-hand column numbers.
GCF = 2 x 3 = 6.
Proceeding to find the LCM, look for any number that goes into two or more of the remaining numbers. One such number is 3, which goes into the remaining 3 and 6. Divide those numbers by 3 and leave the other numbers as they are.
2 | 6 12 18 30 36
3 | 3 6 9 15 18
3 | 1 2 3 5 6
……… 1 2 1 5 2
5th) Interesting! Notice that 2 goes into the two remaining 2s, so pull out a 2 and show the results below:
2 | 6 12 18 30 36
3 | 3 6 9 15 18
3 | 1 2 3 5 6
2 | 1 2 1 5 2
…….. 1 1 1 5 1
6th) We’ve whittled the bottom row’s numbers so far down that finally there’s no number that goes into two or more of them (except 1, which doesn’t help).
So we have all the numbers we need to find the LCM. Multiply them together. The left column gives us: 2 x 3 x 3 x 2. The bottom row gives us 1 x 1 x 1 x 5 x 1. Multiply all of those (non-1) numbers together, you get:
2 x 2 x 3 x 3 x 5 = 180, and that is the LCM! Pretty amazing, huh? And no prime factorizing, to boot.
Some people find that this process takes a bit of practice to get used to it. So here are a few problems to help you become an LCM-finding expert!
a) 12, 18, 30
b) 8, 18, 24
c) 15, 20, 30, 35
d) 16, 24, 40, 56
e) 16, 48, 64, 80, 112
And the answers. LCM for each set is:
a) 180
b) 72
c) 420
d) 1680
e) 6720
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LCM of 3 Numbers Calculator | How to Calculate Least Common Multiple?
The Least Common Multiple is the smallest integer that is divisible by all the numbers. For three integers a, b, c LCM is denoted as LCM(a, b, c). It is also known as LCD or Least Common Denominator.
For instance LCM of 12, 15, 10 is 60 the smallest number that is divisible by all three numbers. Therefore LCM(12, 15, 10) = 60.
Procedure to find the Least Common Multiple of 3 Numbers
Check out the manual procedure on how to find LCM of 3 numbers by hand using different techniques. Each of the methods is explained using step by step process along with solved examples. Understand the concept better and choose the method that is convenient for you. If you feel any difficulty They are along the lines
- Listing Multiples
- Prime Factorization
- Cake/Ladder Method
- Division Method
- Using the Greatest Common Factor GCF
LCM of 3 Numbers using Listing Multiples
- List all the multiples of 3 numbers till you have at least one of the multiples appear on all the lists.
- Find the number that is the smallest and is present on all the lists.
- That itself is the Least Common Multiple of the numbers.
Example
Find LCM(6, 7, 21) using Listing Multiples?
Solution:
Given numbers are 6, 7, 21
Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60.
Multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56
Multiples of 21 are 21, 42, 63
As per the definition, LCM is the smallest number that is common in all 3 numbers multiples.
Thus, 42 is the LCM of 6, 7, 21.
LCM of Three Numbers using Prime Factorization
- Make a note of all the Prime Factors of the given numbers.
- List all the prime numbers found as many times they occur often for the given numbers.
- Multiply the list of prime numbers to obtain the Least Common Multiple.
Example
Find LCM of 12, 24, 30 using Prime Factorization?
Solution:
Prime Factorization of 12 = 2 × 2 × 3
Prime Factorization of 24 = 2 × 2 × 2 × 3
Prime Factorization of 30 = 2 × 3 × 5
Make a note of prime numbers that occur often for the given numbers and multiply them
= 2×2×2×3×5
= 120
How to find LCM of 3 Numbers using the Cake/ Ladder Method?
The cake or Ladder Method is the easiest method to find the Least Common Multiple of all the methods.
Go through the procedure below
- Write down all the 3 numbers in a cake or row.
- Divide the numbers in the layer with a prime number that is evenly divisible in two or more numbers and bring the result into the next layer.
- If any number in the layer or row isn’t divisible just bring it down.
- Continue dividing rows by prime numbers.
- When there are no more prime numbers that divide evenly two or more numbers you are done.
Example
Find the LCM(10, 12, 15) using the Cake/ Ladder Method?
Solution:
Given Numbers are 10, 12, 15
To find LCM Multiply all the prime numbers from top to bottom i.e. 2*3*5*1*2*1
= 60
Therefore, LCM(10, 12, 15) is 60
If you are looking for help on finding the LCM, GCF concepts you can always visit the portal lcmgcf.com to clear all your queries.
Steps to find LCM of 3 Numbers using the Division Method?
- Write down all the numbers in a top table row.
- Begin with the Lowest Prime Numbers and divide the numbers of row by prime numbers that are evenly divisible into at least one of the numbers and bring it to the next row.
- If any number is not divisible bring it down.
- Continue dividing with Prime Numbers that divide at least one number.
- When the last rows are 1’s you are done.
Example
Find LCM of 10, 18, 25 using the Division Method?
Solution:
LCM is the Product of Prime Numbers in the first column
LCM = 2 × 3 × 3 × 5 × 5
LCM = 450
Therefore, LCM(10, 18, 25) is 450.
Finding LCD of 3 Numbers using GCF?
- Initially, find the LCM of the first two numbers.
- Later, find the LCM of the result obtained for the first two numbers and the third number.
- This way, you can find the Least Common Multiple of 3 Numbers easily.
Example
Find the LCM(12, 24, 36) using the GCF Method?
Solution:
Given Numbers are 12, 24, 36
Step 1:Find the LCM of the first 2 numbers 12, 24
LCM of 12, 24 is 24 the smallest number that is divisible by both the numbers.
Step 2: Now take the LCM of the result i.e. 24, 36
LCM(24, 36) = 72
72 is the smallest number that is divisible by both the numbers.
Therefore, the LCM(12, 24, 36) is 72.
1. What does LCM Stand for?
LCM Stands for Least Common Multiple. It is also known as the Least Common Denominator or LCD.
2. What is meant by Least Common Multiple?
The Least Common Multiple is the smallest integer that is divisible by all the numbers.
3. What are the different methods to find LCM?
There are five different methods to find LCM and they are as such
- Listing Multiples
- Prime Factorization
- Cake/Ladder Method
- Division Method
- Using the Greatest Common Factor GCF
4. How to find LCM of 3 numbers using a Calculator?
Simply provide the inputs in the input field and click on the calculate button to get the Least Common Multiple along with detailed work in no time.
90,000 the smallest total multiple algorithm for finding the least common multiple of the largest common divider and the smallest common two natural numbers
| Mathematics Handbook | Arithmetic | ,000 | 9000 9000 9000 |
9001
| Common multiple. Least common multiple |
| Least common multiple algorithm |
| Relationship between the greatest common divisor and the least common multiple of two natural numbers |
Common multiple. Least common multiple
DEFINITION 1. If a natural number a is divisible by a natural number b, then the number a is called a multiple of of the number b.
DEFINITION 2. A common multiple of several natural numbers is a natural number that is a multiple of each of these numbers.
In particular, the common multiple of several numbers is the product of these numbers.
DEFINITION 3. The least common multiple of several natural numbers is called the least common multiple (LCM) of these numbers.
Algorithm for finding the least common multiple
Consider the algorithm for finding the least common multiple of several numbers in the following example.
EXAMPLE. Find the least common multiple of the numbers 100 , 750 , and 800 .
SOLUTION. Let us decompose these numbers into prime factors:
The prime factor 2 enters the power of 2 in the first factorization, the power of 1 in the second factorization, and the power of 5 in the third factorization. Let’s denote the largest of these powers by the letter k .
Obviously, k = 5.
The prime factor 3 enters the first factorization to the power of 0 (in other words, the factor 3 does not enter the first factorization at all), enters the second factorization to the power of 1, and enters the third factorization to the power of 0. Denote the largest of these powers is the letter l. Obviously, l = 1.
The prime factor 5 is included in the first factorization to the power of 2, in the second factorization to the power of 3, and in the third factorization to the power of 2. We denote the largest of these powers by the letter m . Obviously, m = 3.
Now consider the number:
that number is the least common multiple of numbers 100 , 750 , and 800 .
ANSWER: 12000 .
Relationship between the greatest common divisor and the least common multiple of two natural numbers
STATEMENT.
The least common multiple of two numbers can be found by dividing the product of those numbers by their greatest common divisor.
Indeed, consider, for example, two numbers: 10 and 75. Decomposing these numbers into prime factors, we get
Using the algorithms for finding the greatest common divisor and least common multiple of numbers, we find that the greatest common divisor of these numbers is 5, and the least common multiple of these numbers is 150. Since the product of the numbers 10 and 75 is 750, then the relation
which was required to be shown.
NOTE. Since the greatest common divisor of two relatively prime numbers is 1, then the least common multiple of two relatively prime numbers is equal to their product.
How to find the LCM (least common multiple)
A common multiple of two integers is an integer that is evenly divisible without remainder by both given numbers.
The least common multiple of two integers is the smallest of all integers that is evenly and without remainder divisible by both given numbers.
Method 1 . You can find the LCM, in turn, for each of the given numbers, writing out in ascending order all the numbers that are obtained by multiplying them by 1, 2, 3, 4, and so on.
Example for numbers 6 and 9.
We multiply the number 6, sequentially, by 1, 2, 3, 4, 5.
We get: 6, 12, 18 , 24, 30
We multiply the number 9, sequentially, by 1, 2, 3, 4, 5.
We get: 9, 18 , 27, 36, 45
As you can see, the LCM for the numbers 6 and 9will be equal to 18.
This method is convenient when both numbers are small and it is easy to multiply them by a sequence of integers. However, there are cases when you need to find the LCM for two-digit or three-digit numbers, and also when there are three or even more initial numbers.
Method 2 . You can find the LCM by decomposing the original numbers into prime factors.
After decomposition, it is necessary to cross out the same numbers from the resulting series of prime factors. The remaining numbers of the first number will be the factor for the second, and the remaining numbers of the second number will be the factor for the first.
Example for numbers 75 and 60.
The least common multiple of the numbers 75 and 60 can be found without writing out multiples of these numbers in a row. To do this, we decompose 75 and 60 into prime factors:
75 = 3 * 5 * 5, but
60 = 2 * 2 * 3 * 5 .
As you can see, the factors 3 and 5 occur in both rows. Mentally we «cross out» them.
Let’s write down the remaining factors included in the expansion of each of these numbers. When decomposing the number 75, we are left with the number 5, and when decomposing the number 60, we have 2 * 2
So, to determine the LCM for the numbers 75 and 60, we need to multiply the remaining numbers from the expansion of 75 (this is 5) by 60, and the numbers remaining from the expansion of the number 60 (this is 2 * 2) multiply by 75.
That is, for ease of understanding , we say that we multiply «crosswise».
75 * 2 * 2 = 300
60 * 5 = 300
This is how we found the LCM for the numbers 60 and 75. This is the number 300.
Example . Determine LCM for numbers 12, 16, 24
In this case, our actions will be somewhat more complicated. But, first, as always, we decompose all numbers into prime factors
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
24 = 2 * 2 * 2 * 3
To correctly determine the LCM, we select the smallest of all numbers (this is the number 12) and successively go through its factors, crossing them out if at least one of the other rows of numbers has the same factor that has not yet been crossed out.
Step 1. We see that 2 * 2 occurs in all series of numbers. We cross them out.
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
24 = 2 * 2 * 2 * 3
however, for the number 16, no action is expected.
