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Difference between GCD and LCM

Last Updated : 23 Jul, 2025

GCD (Greatest Common Divisor) and LCM (Least Common Multiple) are fundamental mathematical concepts that play a key role in arithmetic and problem-solving. GCD focuses on finding the largest common factor between numbers, LCM identifies the smallest number divisible by them.

GCD

GCD - Greatest Common Divisor, is the largest number that divides two or more numbers without leaving a remainder. In other words, it is the greatest factor common to all the given numbers.

Example : GCD of 24 and 18 s 6 as 6 is the largest number that can evenly divide both of the numbers.

LCM

LCM - Least Common Multiple , is the smallest number that is a multiple of two or more given numbers. It is the smallest number that can be evenly divided by all the given numbers without leaving a remainder.

Example : LCM of 24 and 18 is 72 as 72 is the smallest number that both 12 and 18 can divide evenly

Difference between GCD and LCM

Some of the main differences between GCD and LCM are as follows :

GCD

LCM

It is the Greatest Common Divisor.It is the Least Common Multiple.
The greatest of all the common factors among the given numbers is GCD.The smallest of all the common multiples among the given numbers is LCM.

GCD of two or more numbers is the largest number that divides all of the respective numbers without leaving a remainder.

LCM of two or more numbers is the smallest multiple that is divisible by all the respective numbers.

The GCD of given numbers will be always less than or equal to any of the numbers.The LCM of the given numbers will always be greater than or equal to any of the numbers given.

It is represented as GCD(a, b) where “a” and “b” are two numbers.

It is represented as LCM(a,b) where “a” and “b” are two numbers.

It will involve the identification of common prime factors and multiplying them.

It involves identifying all prime factors and multiplying the maximum occurrence of each factor.

If a = p2 × q3 × r and b = p5 × q2 × r3
then GCD(a, b)= pmin(2, 5) × qmin(3, 2) × rmin(1, 3) = p2 × q2 × r1

If a = p2 × q3 × r and b = p5 × q2 × r3
then LCM(a, b)= pmax(2, 5) × qmax(3, 2) × rmax(1, 3) = p5 × q3 × r3

Used in division, simplifying fractions, and problems involving factors and divisors

Used in multiplication, adding and subtracting fractions, and problems involving multiples and common intervals

Uses of GCD and LCM in Mathematics

Applications of GCD:

  • Simplifying Fractions: GCD is used to reduce fractions to their simplest form by dividing both the numerator and denominator by their GCD.
    Example: Simplify 36/48. GCD of 36 and 48 is 12. Simplified fraction: 3/4.
  • Dividing Items Equally: GCD is useful when dividing items into groups without leftovers.
    Example: Divide 36 apples and 48 oranges into the largest possible equal groups. GCD is 12, so each group will have 12 items.

Applications of LCM:

  • Finding Common Cycles: LCM is used to determine when events that repeat at different intervals will coincide.
    Example: If two traffic lights change every 12 and 18 seconds, they will coincide every 36 seconds (LCM of 12 and 18).
  • Adding Fractions: LCM is used to find a common denominator when adding or subtracting fractions.
    Example: Add .Adding 2/3 and 3/4, LCM of 3 and 4 is 12. Convert fractions: 8/12 and 9/12 now sum of 2/3 and 3/4 = (8 + 9)/12 = 17/12.

Read More : Real life application of GCD and LCM

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