How To Find Lcm Of 2 Numbers
What is Least Common Multiple
The to the lowest degree common multiple is the smallest positive integer that is divisible by both given numbers.
Note:
- If either a or b is 0, LCM is defined to be zero. Information technology is denoted by LCM.
- The LCM of relative prime numbers is the product of those numbers.
How to find LCM
The Least Common Multiple (LCM) of a group of numbers is the smallest number that is a multiple of all the numbers. For example, the LCM of 16 and xx is 80; lxxx is the smallest number that is both a multiple of 16 and a multiple of 20. You can find the LCM of two or more numbers through various methods.
1) Multiple of a Number
This is an ideal method for larger numbers.This method is factoring both numbers downward to the prime numbers that are multiplied to create that number as a product.
For Example :- Permit's say y'all're looking for the least common multiple of 20 and 42. Here'south how you would factor them 20 = 2 x 2 x 5 and 42 = 2 x iii x seven
If the number just occurs in one number, and so it has one occurrence. Here is a list of the most occurrences of each prime number from the previous example two → 2 times iii → one time 5 → i fourth dimension seven → ane time
Since 2 occurs twice, you'll take to multiply it twice. Here's what you should exercise to find the LCM: 2 x ii x 3 x v x 7 = 420.
Video of how to find LCM
Uncomplicated Example
Example 1:
1. What is the LCM of 2 and 3?
6
Explanation:
Find Mutual multiples of 2 and three are just the numbers that are in both lists:
Multiples of 2 are: 2,iv,6,viii,10,12,16,xviii,20
Multiples of iii are: 3,6,ix,12,15,18,21
6,12,xviii……
So the least common multiple of 2 and 3 is the smallest in one of those: 6
Example two:
1. Find the LCM of fifteen and 25.
75
Explanation:
We have to find prime factors that occurs the maximum number of times in any of the numbers.
15 = 5 × iii
25 = 5 × 5
The maximum number of 3 occurs for fifteen. That is 3
The maximum number of 5 occurs for 25. That is 5 × 5
Now practice the product of the prime factors that you establish the maximum number of times in both numbers.
Thus, LCM = 3 × 5 × v = 75
The LCM of 15 and 25 is 75.
Example 3:
ane. Detect the LCM of 20, 28 and 25.
700
Explanation:
The prime factorizations of 20, 28 and 25 are : -
20 = 2 × ii × 5
28 = ii × 2 × vii
25 = 5 × 5
The prime number factor 2 appears maximum number of 2 times in the prime number factorization of 20 and 28. Nosotros will take 2 × 2.
The prime factor 7 occurs one time in the factorization of 28
In the prime factorization of 25, the prime cistron five appears two times.
We will take five × v
At present do the production of the prime factors that you constitute the maximum number of times in each of numbers.
Thus, LCM= 2 × 2 × 5 × 5 × seven = 700
Example four:
1.Observe the LCM of 6, eight.
24
Explanation:
We take to find prime factors that occurs the maximum number of times in whatsoever of the numbers.
6 = ii × iii
8 = 2 × 2 ten 2
The prime factor two appears maximum number of four times in the prime factorization of 6 and eight. We will take two × 2 10 two.
The prime gene three appears maximum number of ane times in the prime factorization of 6 and 8. We volition take 3
Now practice the product of the prime factors that you lot found the maximum number of times in both numbers.
Thus, LCM = ii × iii x 2 × 2 = 24
The LCM of half dozen and 8 is 24.
Example v:
one.Find the LCM of 8 and 12.
24
Explanation:
Nosotros have to observe prime factors that occurs the maximum number of times in any of the numbers.
8 = 2 ten two x 2
12 = two x two ten 3
The prime factor 2 appears maximum number of v times in the prime number factorization of 8 and 12. We will have 2 × two 10 2.
The prime factor 3 appears maximum number of one times in the prime factorization of 8 and 12. We will accept 3
At present do the production of the prime factors that you found the maximum number of times in both numbers.
Thus, LCM = 2 x two 10 2 x 3 = 24
The LCM of viii and 12 is 24.
Instance 6:
1.Find the LCM of 9 and 12.
36
Explanation:
We have to find prime factors that occurs the maximum number of times in whatsoever of the numbers.
9 = iii × three
12 = three × ii x 2
The prime number cistron 3 appears maximum number of 3 times in the prime factorization of 9 and 12. Nosotros will take three x 3.
The prime number gene ii appears maximum number of two times in the prime factorization of ix and 12. We will take two 10 ii
At present practise the product of the prime number factors that you found the maximum number of times in both numbers.
Thus, LCM = 3 × iii x 2 × 2 = 36
The LCM of nine and 12 is 36.
Example 7:
1.Notice the LCM of half dozen and 9.
xviii
Explanation:
We accept to find prime factors that occurs the maximum number of times in whatsoever of the numbers.
6 = 3 × 2
9 = 3 × 3
The prime number factor 3 appears maximum number of three times in the prime factorization of 6 and nine. We volition take 3 x 3.
The prime gene 2 appears maximum number of one times in the prime factorization of 6 and nine. We will take two
At present do the product of the prime factors that you constitute the maximum number of times in both numbers.
Thus, LCM = iii × 3 ten 2 = eighteen
The LCM of 6 and 9 is 18.
two) LCM by Division Method
Write the numbers at the height of the Common Factors Grid (as shown in the example). Leave a small infinite to the left of the numbers and as much space every bit y'all tin below the numbers. Let's say nosotros're working with the numbers 18, 12, and 30. Simply write each number down in its own row at the meridian of the filigree.
Write the everyman common prime factor of the numbers in the space to the left. Just look out for the smallest prime number gene (such as 2, 3, or five) that you tin pull out of all the numbers. They're all even, so you lot tin pull out 2.
Split each of the original numbers by the common prime gene. Write the quotient below each number. Here'south how to practice it:
18/2 = 9, so write 9 below eighteen.
12/2 = 6, then write 6 below 12.
30/ii = xv, and then write 15 below 30.
Repeat the procedure of pulling out and dividing by the lowest prime factor until no more common factors exist. But echo the process from the previous steps using the numbers 9, 6, and 15 this time.
Pull out a 3 from these numbers. three is the lowest prime cistron, or the smallest prime number that is evenly divisible by both numbers. Find the Least Common Multiple of Ii Numbers
Divide all three numbers by 3 and write the result below those numbers. Discover the Least Common Multiple of Two Numbers
9/3 = 3, so write a iii below the nine; 6/3 = ii, so write a two beneath the six; 15/3 = 5 so write a v beneath the fifteen.
If 2 of the numbers still share a prime common factor, and then continue the process until no pair of lesser numbers have a common cistron. In this particular example, you lot're washed.
For case, if the bottom three numbers are 2, 39, and 122, divide 2 and 122 past two leaving the new bottom row as 1, 39, and 61.
Multiply all the numbers of the first column containing the common prime factors with the numbers at the bottoms of all the other columns. This is the LCM. In this instance, the production of the common factor cavalcade is 6 (2 x 3). Multiply 6 by the numbers at the bottoms of the other columns: half-dozen x iii ten two ten 5 = 180.
The LCM of 18, 12, and 30 is 180.
Elementary Case
1.Discover the LCM of ten, 20, and 40 by Division Method.
200
Caption:
If nosotros first divide the numbers 10, 20 and 40 past ii. Nosotros get the quotient 5, 10, twenty.
Keep on dividing the quotient with 2, 3 and 5 respectively until y'all get one in the all row at last.
Now multiply all the divisors to get LCM of given numbers.
Thus, LCM= 2 × ii × ii × 5 × v = 200
LCM of 10, xx < 40 is 200.
LCM : Give-and-take Problems
H2o Tanker Example
Two small containers contain 250 litres and 550 litres of water respectively. Observe the minimum capacity of a tanker which can hold the water from both the containers when used an exact number of times.
2750
Caption :
Nosotros required the tanker that has minimum chapters to concord the h2o from both water containers in an verbal number of times.
The minimum chapters of such a tanker is LCM of 250 and 550.
Nosotros tin can observe it by division method also.
Now multiply all the divisors to become LCM of given numbers.
Thus, LCM= ii × 5 × v × 5 × xi = 2750
LCM of 250 and 550 is 2750.
Therefore, minimum capacity of such a tanker is 2750 litres.
The first container volition fill the tanker past 11 times and the second will past 5 times.
Points to Remember
Quick Tips
- The LCM of more than ii integers is the smallest integer that is divisible by each of them.
- Common multiples are the numbers that are in lists of each numbers given.
Think
- For two prime numbers a, b; can LCM will exist bigger than their product or smaller than their product or to similar to their product?
- Tin can LCM exist ever equal to 1 of the number given out of two numbers x < y? If yes, explain the scenario?
Bones Math Links
Curlicue
Source: https://www.ipracticemath.com/learn/numbersense/lcm
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