Understanding the Context

The Greatest Common Divisor (GCD) of two or more integers is the largest positive number that divides each of them without leaving a remainder. For example, the GCD of 8 and 12 is 4 because 4 is the largest number that divides both. When working with multiple numbers, we find the common divisors of all numbers and identify the greatest one.

Methods to Find the GCD

There are several methods to compute the GCD:

• Prime factorization: Break each number into its prime factors, then identify the common prime factors raised to the lowest powers.
  
• Euclidean Algorithm: A recursive method that uses division with remainders, extremely efficient for large numbers.
  
• Listing divisors: List all divisors of each number and identify the highest common one.

Key Insights

For five numbers like 24, 36, 48, 60, and 72, prime factorization combined with the Euclidean algorithm offers the most systematic and verified approach.

Step-by-Step: Finding the GCD of 24, 36, 48, 60, and 72

Let’s find the GCD step by step using prime factorization and iterative simplification.

Step 1: Prime Factorization of Each Number

Start by expressing each number as a product of prime factors.

Final Thoughts

• 24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3 = 2³ × 3¹
  
• 36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3 = 2² × 3²
  
• 48 = 2 × 24 = 2 × 2 × 12 = 2 × 2 × 2 × 6 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3¹
  
• 60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5 = 2² × 3¹ × 5¹
  
• 72 = 2 × 36 = 2 × 2 × 18 = 2 × 2 × 2 × 9 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²

Step 2: Identify Common Prime Factors

Now, look for all prime factors common to every number and take the lowest exponent for each.

• Common prime factors: 2 and 3
  
  • Lowest power of 2: Found in 36 (2²) and 60 (2²) → smallest is 2²
    
  • Lowest power of 3: Found in all → 3¹

Step 3: Multiply the Common Factors

GCD = (2²) × (3¹) = 4 × 3 = 12