Least Common Multiple(LCM) - Definition, Formula, Examples

Least Common Multiple

The Least Common Multiple (LCM) is one of the most important concepts in mathematics. It assists us in the solving of problems that require a common multiple of two or more numbers. Knowledge of the least common multiple is helpful in adding or subtraction of fractions, comparing ratios, and in problems involving recurring patterns or periods.

LCM informs us of the minimum number which is a multiple of all numbers given. Knowing how to LCM enables students to simplify their problems, develop better number sense, and a solid base on higher topics in math. Students will learn the least common multiple, LCM formula, various ways to solve it, some examples and so on.

Read More: HCF and LCM Formula

What is Least Common Multiple?

The least common multiple of two or more numbers is the smallest positive number that is exactly divisible by each of them. It is also known as the lowest common multiple.

Key Points about Least Common Multiple

• The LCM is always equal to or greater than the largest of the given numbers.

• It helps in finding common denominators while solving fraction problems.

• The LCM is useful in solving real-life problems involving repetition or time intervals.

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Least Common Multiple Formula

The LCM formula provides a simple and quick way to calculate the least common multiple using the greatest common divisor (GCD), also called the highest common factor (HCF).

Formula: LCM(a, b) = (a × b) ÷ GCD(a, b)

Example: Find the LCM of 8 and 12.

• Step 1: Find GCD(8, 12) = 4

• Step 2: Use the formula

• LCM = (8 × 12) ÷ 4 = 96 ÷ 4 = 24

Therefore, the least common multiple of 8 and 12 is 24.

This LCM formula is very helpful when working with large numbers or algebraic expressions because it saves time and reduces calculation errors.

Read More: Pythagorean Triples

How to Find Least Common Multiple?

Finding the Least Common Multiple (LCM) can be done using different methods. Each method works best depending on the numbers you have. The following are the main ways to find LCM:

1. LCM by Listing Method

The listing method is simple and best for small numbers.

Steps:

1. Write the multiples of each number.

2. Identify the smallest number that appears in all lists.

3. That number is the LCM.

Example: Find LCM of 3 and 5.

• Multiples of 3: 3, 6, 9, 12, 15…

• Multiples of 5: 5, 10, 15, 20…

The smallest common multiple is 15. LCM(3, 5) = 15

This method is also called LCM by listing method.

2. LCM by Prime Factorization

Prime factorization is more systematic and works for larger numbers.

Steps:

1. Write each number as a product of prime factors.

2. Take the highest power of each prime factor.

3. Multiply these primes together to get the LCM.

Example: Find LCM of 18 and 24.

• 18 = 2 × 3²

• 24 = 2³ × 3

Take the highest powers: 2³ × 3² = 8 × 9 = 72. LCM(18, 24) = 72

This method is also called LCM by prime factorization method.

3. LCM by Division Method

The division method is helpful when finding the LCM of two or more numbers.

Steps:

1. Write all numbers in a row.

2. Divide them by a common prime number.

3. Repeat until all numbers become 1.

4. Multiply all the divisors to get the LCM.

Example: Find LCM of 12 and 15.

Step 1: Write 12 and 15 side by side.

Step 2: Divide by 3 (common factor).
12 ÷ 3 = 4, 15 ÷ 3 = 5

Step 3: Divide by 2, 5 cannot be divided by 2, so leave it as it is.
4 ÷ 2 = 2, 5 stays as 5

Step 4: Divide by 2 again.
2 ÷ 2 = 1, 5 stays as 5

Step 5: Divide by 5.
5 ÷ 5 = 1

Now multiply all divisors used: 3 × 2 × 2 × 5 = 60
Hence, LCM(12, 15) = 60.

The LCM by division method is very efficient when finding the least common multiple of three or more numbers.

Read More: GCF (Greatest Common Factor)

Relationship Between LCM and HCF

The relationship between LCM and HCF is very important in mathematics. It helps verify calculations and understand how numbers relate to each other.

Formula: LCM(a, b) × HCF(a, b) = a × b

Example: For a = 8 and b = 12:

HCF(8, 12) = 4

LCM(8, 12) = 24

Now, 4 × 24 = 96 and 8 × 12 = 96.

Both sides are equal.

This shows the strong connection between the LCM and HCF and helps in checking our results easily.

Read More: Division

Difference Between LCM and HCF

It is important to understand the difference between LCM and HCF to use them correctly in different types of problems. The table provides the difference between HCF and LCM:

Knowing the difference between LCM and HCF helps students understand when to use which method for solving math problems correctly.

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