Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store

LCM of 15 and 20 with Step-by-Step Solutions

ffImage
hightlight icon
highlight icon
highlight icon
share icon
copy icon
SearchIcon

How to Find the LCM of 15 and 20 Using Prime Factorization

The concept of LCM of 15 and 20 is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Knowing the Lowest Common Multiple (LCM) makes calculations involving fractions, schedules, and problem-solving much clearer and faster for students.


Understanding LCM of 15 and 20

The LCM of 15 and 20 refers to the smallest positive integer that is exactly divisible by both 15 and 20 without any remainder. This concept is widely used in arithmetic, fractions, and time problems. Calculating LCM is important for adding and subtracting fractions, finding common timings, and solving word problems that involve grouping or repetition.


Formula Used in LCM of 15 and 20

The standard formula is: \( \text{LCM}(A, B) = \dfrac{A \times B}{\text{GCF}(A, B)} \)


Here’s another way, using prime factorization for LCM of 15 and 20:

15 = 3 × 5
20 = 2 × 2 × 5

Take the highest power of every prime number present in both numbers:
LCM (15, 20) = 2² × 3 × 5 = 60


Here’s a helpful table to understand LCM of 15 and 20 more clearly:


LCM of 15 and 20 Table

Number Is 60 Divisible? Result
15 60 ÷ 15 = 4 Yes
20 60 ÷ 20 = 3 Yes
Any number less than 60 Not divisible by both No

This table shows that 60 is the lowest common multiple of 15 and 20—it is divisible by both numbers without remainder, and no smaller number fits this condition.


Worked Example – Solving LCM of 15 and 20

Let’s solve the question step by step using two methods:

Method 1: Prime Factorization Method
1. Find the prime factors of 15 and 20.

15 = 3 × 5
20 = 2 × 2 × 5

2. List all prime numbers present using the highest power from both:

2² (from 20), 3 (from 15), and 5 (common in both, use once)

3. Multiply these prime factors together:

2 × 2 × 3 × 5 = 4 × 3 × 5 = 12 × 5 = 60

Method 2: Division (Continuous Division Method)
1. Write 15 and 20 side by side.

2. Divide both by the smallest prime that divides at least one (start with 2):
  2 | 15, 20 → 15, 10
  2 | 15, 10 → 15, 5
  3 | 15, 5 → 5, 5
  5 | 5, 5 → 1, 1

3. Multiply all divisors used: 2 × 2 × 3 × 5 = 60

Practice Problems

  • Find the LCM of 15, 20, and 30.
  • List all multiples of 15 and 20 up to 100 and find their common multiples.
  • If a bell rings every 15 minutes and another every 20 minutes, after how many minutes will they ring together?
  • Is 120 a common multiple of 15 and 20? Is it the lowest one?

Common Mistakes to Avoid

  • Confusing LCM of 15 and 20 with HCF (Highest Common Factor).
  • Forgetting to use the highest power of each prime when using the prime factorization method for LCM.
  • Stopping at any common multiple, not checking if it is the lowest.

LCM of 15 and 20 vs. HCF of 15 and 20

Property LCM of 15 and 20 HCF of 15 and 20
Full Form Lowest Common Multiple Highest Common Factor
Definition Smallest number divisible by both 15, 20 Greatest number that divides both 15, 20
Value 60 5

This comparison helps avoid confusion between LCM and HCF in exams and problem-solving.


Real-World Applications

The concept of LCM of 15 and 20 is useful for tasks like finding the first time two events coincide, grouping items for packaging, and coordinating schedules in banking or transportation. Vedantu helps students recognize how LCM problems are relevant beyond school subjects.


More About Factors and Multiples

To deepen your understanding of factors and multiples used in LCM methods, check these resources:


We explored the idea of LCM of 15 and 20, how to find it step by step, how to avoid mistakes, and where it applies in daily life. Practice these concepts with Vedantu and strengthen your maths foundation for school and beyond.


FAQs on LCM of 15 and 20 with Step-by-Step Solutions

1. What is the LCM of 15 and 20?

The LCM (Least Common Multiple) of 15 and 20 is 60. It is the smallest number that is divisible by both 15 and 20 without leaving a remainder.

2. How to calculate the LCM of 15 and 20 using prime factorization?

To find the LCM of 15 and 20 by prime factorization, follow these steps:
1. Express each number as a product of its prime factors:
    15 = 3 × 5
    20 = 2 × 2 × 5
2. Take the highest power of all primes that appear: 2², 3¹, 5¹.
3. Multiply them: 2 × 2 × 3 × 5 = 60.
Hence, the LCM is 60.

3. What is the least common multiple of 15 and 20?

The least common multiple of 15 and 20 is the same as their LCM, which is 60. It is the smallest number that both 15 and 20 exactly divide into.

4. How to find LCM of 15, 20 and 30?

To find the LCM of 15, 20, and 30, use prime factorization or division method:
Prime factors:
15 = 3 × 5
20 = 2 × 2 × 5
30 = 2 × 3 × 5
Take highest powers: 2², 3¹, 5¹.
Multiply: 2 × 2 × 3 × 5 = 60. Thus, the LCM is 60.

5. What is the LCM and GCF of 15 and 20?

The LCM of 15 and 20 is 60, while the GCF (Greatest Common Factor) or HCF (Highest Common Factor) of 15 and 20 is 5.
LCM is the smallest number divisible by both numbers, whereas GCF is the largest number that divides both.

6. Why is the LCM of 15 and 20 not 30 or 45?

LCM must be divisible by both numbers. While 30 is divisible by 15, it is not divisible by 20. Similarly, 45 is divisible by 15 but not by 20. The smallest number divisible by both 15 and 20 is 60, making it the correct LCM.

7. Why do students often confuse LCM with HCF?

Students confuse LCM and HCF because both involve common multiples or factors. Key differences are:
- LCM is about common multiples, the smallest number divisible by given numbers.
- HCF is the greatest number that divides the given numbers.
Understanding this distinction helps avoid confusion.

8. How can LCM questions appear in CBSE board exams or Olympiads?

LCM problems in exams often test:
- Basic calculation of LCM using prime factorization or division
- Finding LCM of three or more numbers
- Word problems involving time, speed, or event synchronization
Preparing these types enhances problem-solving skills aligned with CBSE and Olympiad standards.

9. What mistakes should I avoid in prime factorization?

Common mistakes when using prime factorization include:
- Missing prime factors
- Incorrectly factoring composite numbers
- Not using the highest powers of each prime in LCM calculation
Double-check each step to ensure accuracy in factors and powers.

10. Why is the LCM sometimes called the least common denominator?

The LCM is often called the least common denominator in fractions because it represents the smallest shared denominator that can be used to add or compare fractions with different denominators. It simplifies calculations by providing a common base.