askvity

What is LCM and GCM?

Published in Number Theory 3 mins read

The terms LCM and GCM refer to fundamental concepts in number theory related to multiples and factors of integers. Let's define them clearly:

Understanding LCM and GCM

Term Definition
LCM (Least Common Multiple) The smallest positive whole number that is a multiple of two or more given whole numbers.
GCM (Greatest Common Factor) also known as GCF The largest whole number that divides two or more given numbers without leaving a remainder.

Deep Dive into LCM

The Least Common Multiple (LCM) helps when you need to find a common point in situations involving different intervals or cycles.

How to find the LCM:

  • List multiples: Write out multiples of each number until you find a common one.
  • Example: LCM of 4 and 6
  • Multiples of 4: 4, 8, 12, 16, 20, 24,...
  • Multiples of 6: 6, 12, 18, 24, 30,...
  • The LCM of 4 and 6 is 12.
  • Prime factorization: Break each number into its prime factors and multiply together the highest powers of all primes involved.
    • Example: LCM of 12 and 18
      • 12 = 22 * 3
      • 18 = 2 * 32
      • LCM = 22 32 = 4 9 = 36
  • Use the formula: For two numbers, a and b, LCM(a,b) = (|a*b|)/GCD(a,b)

Practical Use Cases of LCM

  • Scheduling: Finding when events will coincide if they happen at different regular intervals.
  • Fractions: Finding the least common denominator when adding or subtracting fractions.

In-Depth Look at GCM (or GCF)

The Greatest Common Factor (GCM) or Greatest Common Divisor (GCD), helps determine the largest number that can divide two or more numbers evenly.

How to find the GCM/GCD:

  • List factors: List the factors of each number and identify the greatest one they share.
    • Example: GCM of 12 and 18
      • Factors of 12: 1, 2, 3, 4, 6, 12
      • Factors of 18: 1, 2, 3, 6, 9, 18
      • The GCM of 12 and 18 is 6.
  • Prime factorization: Break each number into its prime factors and multiply the common primes together, using the lowest powers.
    • Example: GCM of 36 and 48
      • 36 = 22 * 32
      • 48 = 24 * 3
      • GCM = 22 3 = 4 3 = 12
  • Euclidean algorithm: Repeatedly apply the division algorithm until you get a remainder of 0. The last non-zero remainder is the GCM.

Practical Use Cases of GCM

  • Simplifying fractions: Dividing the numerator and denominator by the GCM can simplify the fraction.
  • Splitting items into equal groups: To split a set of items into the largest possible equal groups.

Key Differences

  • The LCM focuses on multiples, the GCM on factors.
  • The LCM is equal to or greater than the input numbers, whereas the GCM is equal to or less than the input numbers.

Related Articles