Facebook Pixel
Mathos AI logo

GCF and LCM

Greatest Common Factor (GCF) and Least Common Multiple (LCM)

Understanding how to find the Greatest Common Factor (GCF) and Least Common Multiple (LCM) is a fundamental skill in number theory. These concepts help us simplify fractions, find common denominators, and solve real-world problems involving repeating events or dividing items equally.

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest number that divides evenly into two or more numbers.

Method 1: Listing Factors

To find the GCF of 36 and 48, list all the factors of each number:

  • Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  • Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

The common factors are 1, 2, 3, 4, 6, and 12. The largest of these is 12, so the GCF is 12.

Least Common Multiple (LCM)

The Least Common Multiple is the smallest positive number that is a multiple of two or more numbers.

Method 1: Listing Multiples

To find the LCM of 12 and 18, list the first few multiples of each number:

  • Multiples of 12: 12, 24, 36, 48, 60...
  • Multiples of 18: 18, 36, 54, 72...

The smallest multiple they share is 36, so the LCM is 36.

Prime Factorization Method

You can find both the GCF and LCM using prime factorization. Let's use 36 and 48 as an example.

First, find the prime factorization of each number:

  • 36=22×3236 = 2^2 \times 3^2
  • 48=24×3148 = 2^4 \times 3^1

For the GCF: Multiply the lowest powers of all common prime factors. GCF=22×31=4×3=12GCF = 2^2 \times 3^1 = 4 \times 3 = 12

For the LCM: Multiply the highest powers of all prime factors present. LCM=24×32=16×9=144LCM = 2^4 \times 3^2 = 16 \times 9 = 144

The Ladder Method

The ladder method (or upside-down division) is a fast way to find both the GCF and LCM at the same time. Let's find the GCF and LCM of 12 and 18.

  1. Write the numbers in an upside-down division bracket.
  2. Divide by the smallest common prime number (like 2).
    • 12÷2=612 \div 2 = 6
    • 18÷2=918 \div 2 = 9
  3. Divide the new numbers (6 and 9) by their common prime factor (3).
    • 6÷3=26 \div 3 = 2
    • 9÷3=39 \div 3 = 3
  4. Stop when the bottom numbers share no common factors other than 1.

To find the GCF: Multiply the numbers on the left side of the ladder. GCF=2×3=6GCF = 2 \times 3 = 6

To find the LCM: Multiply the numbers on the left side AND the bottom of the ladder (creating an "L" shape). LCM=2×3×2×3=36LCM = 2 \times 3 \times 2 \times 3 = 36