The Least Common Multiple (LCM), in the world of integers, is a very important concept that refers to the smallest positive integer that can divided by each of two or more given numbers. LCM plays a key role in solving problems related to adding and subtracting fractions, finding common denominators, and analyzing of recurring events.

Further in this article, we’ll see is Least Common Multiple (LCM), How it is calculated, and with the help of some practice problems designed to help you understand it better.

What is Least Common Multiple (LCM)?

The LCM is the smallest positive integer that is a multiple of all the given integers.

The LCM is the smallest number that all given integers can divide without leftovers. It’s super useful for stuff like adding fractions or finding out when events sync up.

For Example: Take 4 and 5. Their LCM is 20 because it’s the smallest number divisible by both without a remainder.

Important Formulas Related to LCM

1. Prime Factorization:

Break down each number into its prime factors (the basic building blocks of any number). Then, pick the highest power of each prime number found in either number and multiply them together.

2. Using Greatest Common Divisor (GCD):

Alternatively, divide the product of the two numbers by their GCD (the biggest number that divides both evenly). This gives you the LCM. This can be expressed mathematically as:

[Tex]\text{LCM}(a, b) = \frac{|a \cdot b|}{\text{GCD}(a, b)} [/Tex]

Practice Questions on Least Common Multiple (LCM): Solved

1. Calculate the LCM of 12 and 18.

The prime factorization of 12 is 2² × 3, and of 18 is 2 × 3². LCM is 2² × 3² = 36.

2. Calculate the LCM of 15, 20, and 25.

Solution: Prime factorization of 15 is 3 × 5, 20 is 2² × 5, and 25 is 5². LCM is 2² × 3 × 5² = 100.

3. Calculate the LCM of 8, 12, and 18.

Solution: Prime factorization of 8 is 2³, 12 is 2² × 3, and 18 is 2 × 3². LCM is 2³ × 3² = 72.

4. Calculate the LCM of 21, 35, and 49.

Solution: Prime factorization of 21 is 3×7, 35 is 5 × 7, and 49 is 7². LCM is 3 × 5 × 7² = 735.

5. Calculate the LCM of 18 and 24.

Solution: Prime factorization of 18 is 2 × 3² , and of 24 is 2³ × 3. LCM is 2³ × 3² = 72.

6. Calculate the LCM of 16, 24, and 36.

Solution: Prime factorization of 16 is 2⁴, 24 is 2³ × 3, and 36 is 2² × 3². LCM is 2⁴ × 3² = 144.

7. Calculate the LCM of 42, 56, and 84.

Solution: Prime factorization of 42 is 2 × 3 × 7, 56 is 2³ × 7, and 84 is 2² × 3 × 7². LCM is 2³ × 3 × 7 = 168.

8. Calculate the LCM of 9, 15, and 25.

Solution: Prime factorization of 9 is 3², 15 is 3 × 5, and 25 is 5². LCM is 3² × 5² = 225.

9. Calculate the LCM of 20, 30, and 40.

Solution: Prime factorization of 20 is 2² × 5, 30 is 2 × 3 × 5, and 40 is 2³ × 5. LCM is 2³ × 3 × 5 = 120.

10. Calculate the LCM of 36 and 48.

Solution: Prime factorization of 36 is 2² × 3², and of 48 is 2⁴ × 3. LCM is 2⁴ × 3² = 144.

Practice Questions on Least Common Multiple (LCM): Unsolved

1. Find the LCM of 14 and 21.

2. Find the LCM of 27, 36, and 45.

3. Find the LCM of 16 and 24.

4. Find the LCM of 54 and 72.

5. Find the LCM of 8, 12, and 20.

6. Find the LCM of 63 and 84.

7. Find the LCM of 30 and 45.

8. Find the LCM of 18, 24, and 36.

9. Find the LCM of 42 and 56.

10. Find the LCM of 15, 25, and 35.

FAQs on Least Common Multiple (LCM)

What is the Least Common Multiple?

The LCM is the smallest positive integer that is a multiple of all the given integers.

Methods to find the LCM of two numbers?

Few methods to find LCM: Prime Factorization, Division Method, Listing Multiples

What is the relationship between LCM and GCD?

LCM(a,b) x GCD(a,b) = a x b

Can the LCM be zero?

No.