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What is the Least Common Multiple (LCM)?
The least common multiple (LCM) is the smallest number that is divisible by each of two or more numbers. The LCM can be found by listing the multiples of each number and finding the smallest number that is listed multiple times. For example, the LCM of 12 and 18 is 36 because the multiples of 12 are 12, 24, 36, 48, and 60, and the multiples of 18 are 18, 36, 54, 72, and 90. The LCM of 15 and 30 is 60 because the multiples of 15 are 15, 30, 45, 60, and 75, and the multiples of 30 are 30, 60, 90, 120, and 150.
How to Find LCM of Two Numbers?
There are 4 main methods to calculate the least common multiple of 2 numbers. These methods are:
- Listing Multiples or Brute Force Method
- Prime Factorization Method
- Division Method or Ladder Method
- GCD or GCF Method
Listing Multiples or Brute Force Method of Finding LCM
In this method, the multiples of each number are listed until the first common multiple is found. Consider the example of 4 and 10. For this method, the multiples of 4 and 10 are to be listed.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28
Multiples of 10: 10, 20, 30, 40
Here, the number 20 is the first common multiple of both 4 and 10. So, the LCM of 4, 10 is 20.
Prime Factorization Method of Finding LCM
The prime factorization is one of the most common ways of finding LCM. To find the LCM of two numbers 30 and 45, the steps are as follows:
Step 1: To first list the prime factors of each number.
30 = 2 × 3 × 5
45 = 3 × 3 × 5
Step 2: Next multiply each factor the maximum number of times it occurs in either number.
If the same factor occurs more than once in both numbers, then multiply the factor the maximum number of times it occurs.
The occurrence of Numbers in the above example:
2: one time
3: two times
5: one time
LCM = 2 × 3 × 3 × 5 = 90
After calculating the LCM, always check to be sure your answer can be divided evenly by both numbers.
Division Method or Ladder Method of Finding LCM
In this method, the two numbers are simultaneously divided with prime numbers until the division is even. When there are no more primes that evenly divide into both numbers, multiply the divisors to get the LCM. For example, consider 4 and 10 as two
Division Method or Ladder Method of Finding LCM
Here, the LCM of 24, 15 will be 2×2×2×3×5=23×3×5=120
GCD or GCF Method of Finding LCM
This method is used only when the greatest common factor of two numbers is given. The formula used to find the LCM using the GCF or GCD is:
L.C.M. = a×b/ gcd(a,b)
For example, for 15 and 24, the GCF will be 3. So, the LCM will be (15 × 24) / 3 = 3.
Solved Examples
Example 1: Find the L.C.M of 18 and 24 by using the division method?
Solution:
| 2 | 18 | 24 |
| 2 | 9 | 12 |
| 2 | 9 | 6 |
| 3 | 9 | 3 |
| 3 | 3 | 1 |
| 1 | 1 |
For numbers 18 and 24 = 2 × 2 × 2 × 3 × 3 = 72 is the LCM.
Example 2: Find the Least Common Multiples of these sets of numbers: 3, 9, 21.
Solution:
Step 1: List the prime factors of each.
3: 3
9: 3 × 3
21: 3 × 7
Step 2: Multiply each factor the maximum number of times it occurs in any of the numbers.
The occurrence of Numbers in the above example:
3: two times
7: one time
3 x 3 x 7 = 63
9 has two 3s, and 21 has one 7, so we multiply 3 two times, and 7 once.
This gives us 63, the lowest number that can be divided evenly by 3, 9, and 21.
Example 3: Find the Least common factor of 12, 80.
Solution:
Step 1: List the prime factors of each.
12: 2 × 2 × 3
80: 2 × 2 × 2 × 2 × 5
Step 2:
Multiply each factor the maximum number of times it occurs in either number.
Step 3:
The occurrence of Numbers in the above example:
2: 4 times
3: 1 time
5: 1 time
2 x 2 x 2 x 2 x 3 x 5 =240
12 has one 3, and 80 has four 2’s and one 5, so we multiply 2 four times, 3 once, and five once.
This gives us 240, the lowest number that can be divided by both 12 and 80.
Difference between GCF & LCM
GCF is the Greatest Common Factor and LCM is the Least Common Multiple.
GCF is the largest number that is a factor of both numbers. LCM is the smallest number that is a multiple of both numbers.
