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LCM of two numbers is the smallest common multiple that is fully divided by both numbers.
- You can define a common multiple as a number that is a multiple of two or more numbers.
- Therefore, LCM is the least common multiple between two or more numbers that are completely divisible by them.
- For example, if the LCM of x and y is equal to z, then z should be divisible by both x and y.
In this article, you will get the easiest methods to find the LCM of two numbers and some solved examples that make understanding the concept clear and straightforward.
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Key Terms: LCM, HCF, Least common factor, Fractions, Prime factorization, Greatest common factor, GCD, Division, Multiplication
What is LCM of 2 Numbers?
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In mathematics, LCM of 2 numbers is defined as the smallest number that is divided by both numbers evenly. LCM is also known as the Least Common Divisor (LCD).
For example, if we consider 2 numbers i.e. 20 and 5.
- The smallest number, or the least number that can be divided by 20 and 5 evenly, will be 20.
- Therefore, we can say that the LCM of 20 and 5 is 20.
- We can also find the LCM using the prime factorization method.
Also check:
| Related Concepts | ||
|---|---|---|
| Highest Common Factor | Relation between HCF and LCM | Euclid’s Division Lemma |
| Prime Factorization of HCF and LCM | Properties of LCM and HCF | Denominator |
How to Find LCM of Two Numbers?
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There are many different methods but here we will discuss the four main methods to find the LCM of 2 Numbers. The methods are as follows:
- Listing Multiples or Brute Force Method
- Prime Factorization Method
- Division Method or Ladder Method
- GCD or GCF Method
Listing Multiples or Brute Force Method
In this method, we simply list the multiples of each number until we find a common multiple they share.
Listing Multiples Method ExampleQues. Find the LCM of 3 and 6 using the listing multiples method. Ans. As per the listing method multiples of 3 and 9 are to listed that are:
9 is the first common multiple of 3 and 9. Therefore, their LCM is 9. |
Prime Factorization Method
It is one of the most common methods of finding the LCM of two numbers.
Prime Factorization Method ExampleQues. Find the LCM of 12 and 18 using the prime factorization method. Ans. The steps of the prime factorization method to find LCM of 12 and 18 are: Step 1: We start by finding the prime factorization of each number.
Step 2: Now, we multiply each factor by the maximum number of times. The number of times a number occurred in the prime factorization of 12 and 18.
LCM = 2 x 2 x 3 x 3 = 36 It must be noted that, after calculating the LCM, we must always check whether the answer is divided evenly by both numbers. |
Division Method or Ladder Method
In the division method, you can find the LCM of two numbers by dividing them by prime numbers until the division is even. When no prime number divides both the numbers then you multiply the divisor to get the LCM.
Division Method ExampleQues. Find the LCM of 24 and 15 using the division method. Ans. The two given numbers are 24 and 15  LCM of 24 and 15 Therefore, the LCM of 24 and 15 will be 2 x 2 x 2 x 3 x 5 = 120 |
GCD or GCF Method
In case, when the greatest common factor (GCF) of two numbers is given, we use the GCF method to find the LCM.
The LCM formula of 2 numbers by GCF method is:
\(LCM = \frac {a\: \times\: b}{GCF\:(a,\:b)}\)
GCF or GCD Method ExampleQues. Find the LCM of two numbers 15 and 24 using the GCF method. Ans.Given
The greatest common factor of 15 and 24 is, GCF = 3 The formula of the relation between GCF and LCM of two numbers is LCM = (a x b) / (GCF) On substituting the values, we get LCM = (15 x 24) / 3 = 360 /3 ⇒ LCM = 120 Therefore, the LCM of 15 and 24 is 120. |
How to Find LCM of Fractions?
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The LCM of two fractions is the least common multiple that can be divided completely by the two fractions. The formula to find the LCM of two fractions is given by
\(L.C.M = \frac {LCM \: of \: numerators}{HCF \: of \: denominators}\)
Let p/q and r/s be two fractions. The following steps are used to find the LCM of p/q and r/s
- Step 1: Find the LCM of the numerator i.e. LCM (p, r)
- Step 2: Find the HCF of the denominator i.e. HCF (q, s)
- Step 3: Put the values in the given formula.
LCM of Fractions ExampleQues. Find the LCM of 2/3 and 6/7. Ans. Given
Step 1: The LCM of 2 and 6 is 6. Step 2: The HCF of 3 and 7 is 1. Step 3: Substituting the obtained values in the formula. The formula to find the LCM of two fractions is given by LCM = (LCM of numerators) / (HCF of denominators) ⇒ LCM = 6 / 1 ⇒ LCM = 6 Therefore, the LCM of fractions 2/3 and 6/7 is 6. |
Solved Examples
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| Ques. Find the LCM of 16 and 18 using the prime factorization method. Ans. The steps of the prime factorization method to find LCM of 12 and 18 are: Prime factorization of each number.
The number of times a number occurred in the prime factorization of 12 and 18.
Multiply each factor the maximum number of times. LCM = 2 x 2 x 2 x 2 x 3 x 3 = 144 Ques. Find the LCM of two numbers 18 and 30 using the GCF method. Ans.Given
The greatest common factor of 18 and 30 is, GCF = 6 The formula of the relation between GCF and LCM of two numbers is LCM = (a x b) / (GCF) On substituting the values, we get LCM = (18 x 30) / 6 = 540 / 6 ⇒ LCM = 90 |
Also check:
| Related Concepts | ||
|---|---|---|
| Congruence of Triangles | Pythagoras Theorem | Similarity of Triangles |
| Area of Parallelogram | Linear Equations in One Variable | Rational Numbers |
Things to Remember
- The full form of LCM is the least common multiple.
- The LCM of two or more numbers is the smallest common multiple of two numbers.
- There are four methods to find LCM: Listing method, prime factorization method, division method, and GCF method.
- The formula to find LCM using the GCF method is LCM = (a x b) / (GCF).
- The formula to find the LCM of two fractions is LCM = (LCM of numerators) / (HCF of denominators).
Sample Questions
Ques. Find the LCM of 9 and 15 using the listing multiples method. (2 Marks)
Ans. As per the listing method multiples of 3 and 9 are to listed that are:
- Multiple of 9: 9, 18, 27, 36, 45…….
- Multiple of 15: 15, 30, 45, 60…
45 is the first common multiple of 3 and 9. Therefore, their LCM is 45.
Ques. Find the LCM of two numbers 9 and 45 using the GCF method. (2 Marks)
Ans.Given
- a = 9
- b = 45
The greatest common factor of 9 and 45 is, GCF = 9
The formula of the relation between GCF and LCM of two numbers is
LCM = (a x b) / (GCF)
On substituting the values, we get
LCM = (9 x 45) / 9 = 405 / 9
⇒ LCM = 45
Ques. What is the LCM of 2 and 4? (2 Marks)
Ans. As per the listing method multiples of 3 and 9 are to listed that are:
- Multiple of 2: 2, 4, 6, 8, 10…….
- Multiple of 4: 4, 8, 12, 16…….
4 is the first common multiple of 2 and 4. Therefore, their LCM is 4.
Ques. Who invented the concept of LCM? (1 Mark)
Ans. The concept of Least Common Factor (LCM) was invented by the mathematician Euclid.
Ques. What is the definition of LCM of 2 numbers? (1 Mark)
Ans. LCM of 2 numbers is the smallest common multiple that divides both numbers.
Ques. What are the different methods of finding LCM of 2 numbers? (2 Marks)
Ans. There are a total of 4 methods to find the LCM of 2 numbers:
- Listing Multiples or Brute Force Method
- Prime Factorization Method
- Division Method or Ladder Method
- GCF Method
Ques. What is the formula of the GCF Method to find the LCM of two numbers? (1 Mark)
Ans. The formula used to find the LCM of two numbers through the GCF method is LCM = Multiplication of Two Numbers/GCF
Ques. How to find the LCM of two numbers that have nothing in common? (2 Marks)
Ans. In situations where the two numbers have nothing in common the method to find the LCM is multiplying the numbers with each other. For example, 5 and 7 are the two numbers. They have nothing in common. So, the LCM of 5 and 7 is 5x7 = 35
Ques. Find the LCM of 18 and 24 by using the listing method. (2 Marks)
Ans. Multiple of 18 are 18, 36, 54, 72, 90, 108….
Multiple of 24 are 24, 48, 72, 96, 120…
Thus, the LCM of 18 and 24 is 72.
Ques. Find the LCM of 12 and 18 through the prime factorization method. (3 Marks)
Ans.
Step 1: Listing the prime factors of each number.
12: 2 × 2 × 3
80: 2 × 2 × 2 × 2 × 5
Step 2: Multiply each factor by the maximum number of times it occurs in either number.
Step 3: Numbers occurrence
2: 4 times
3: 1 time
5: 1 time
LCM of 12 and 80 is 2 x 2 x 2 x 2 x 3 x 5 =240
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