Practice Questions on Least Common Multiple ( LCM ) - Basic

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. In other words the LCM is the smallest positive integer that is a multiple of all the given integers.

Example: The LCM of 4 and 5 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:

\text{LCM}(a, b) = \frac{|a \cdot b|}{\text{GCD}(a, b)}

Solved Examples on LCM ( Least Common Multiple ) - Basic

Question 1. Calculate the LCM of 12 and 18.

Solution:

Prime factorization of 12 is 2² × 3,
Prime factorization of 18 is 2 × 3²

LCM is 2² × 3² = 36.

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

Solution:

Prime factorization of 15 is 3 × 5
Prime factorization of 20 is 2² × 5
Prime factorization of 25 is 5²

LCM is 2² × 3 × 5² = 100.

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

Solution:

Prime factorization of 8 is 2³
Prime factorization of 12 is 2² × 3
Prime factorization of 18 is 2 × 3²

LCM is 2³ × 3² = 72.

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

Solution:

Prime factorization of 21 is 3 × 7
Prime factorization of 35 is 5 × 7
Prime factorization of 49 is 7²

LCM is 3 × 5 × 7² = 735.

Question 5. Calculate the LCM of 18 and 24.

Solution:

Prime factorization of 18 is 2 × 3²
Prime factorization of 24 is 2³ × 3

LCM is 2³ × 3² = 72.

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

Solution:

Prime factorization of 16 is 2⁴
Prime factorization of 24 is 2³ × 3
Prime factorization of 36 is 2² × 3²

LCM is 2⁴ × 3² = 144.

Question 7. Calculate the LCM of 4y, 5y, and 8z, where z = 2y.

Solution:

Prime factorization of 4y is 2² × z
Prime factorization of 56 is 5 × z
Prime factorization of 84 is 2³ × y = 2³ × 2 × z = 2⁴ × z

LCM is 2⁴ × 5 × z = 40z.

Question 8. Calculate the LCM of 9, 18, and 36.

Solution:

As we know LCM ( a, b, c) = LCM( a, LCM(b, c))
LCM( 9, 18, 36) = LCM( 9, LCM(18, 36))

as 36 is multiple of 18 LCM( 18, 36 ) = 36
Also , 36 is multiple of 9 then LCM ( 9, 36) = 36

LCM of 9, 18, and 36 is 36

Question 9. Calculate the LCM of 2y, 3y, and 4y.

Solution:

Prime factorization of 2y is 2 × y
Prime factorization of 30 is 3 × y
Prime factorization of 40 is 2²× y

LCM is 2²× 3 × y = 12y.

Question 10. Find the smallest number which when divided by 15, 25 and 50 leaves a remainder of 5.

Solution:

Let the number be x
then,
x - 5 = smallest number which can be evenly divided by 15, 25 and 50
x- 5 = LCM ( 15, 25, 50)
x = LCM ( 15, 25, 50) + 5 ------ (i)

Prime factorization of 15 is 3 × 5
Prime factorization of 25 is 5²
Prime factorization of 50 is 5² × 2

LCM( 15, 25, 50) = 3 × 5² × 2
LCM( 15, 25, 50) = 150

Putting the value in (i)

x = 150 +5
x = 155

Next Article - Practice Questions on Least Common Multiple ( LCM ) - Advanced

Practice Questions on Least Common Multiple (LCM): Unsolved

Question 1. Find the LCM of 14 and 21.

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

Question 3. Find the LCM of 16 and 24.

Question 4. Find the LCM of 54 and 72.

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

Question 6. Find the LCM of 63 and 84.

Question 7. Find the LCM of 30 and 45.

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

Question 9. Find the LCM of 42 and 56.

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

Answer Key

1. 42
2. 540
3. 48
4. 216
5. 120
6. 252
7. 90
8. 72
9. 168
10. 525