Understanding the Context

The GCD of two integers is the largest number that divides both without leaving a remainder. Prime factorization breaks each number into its fundamental building blocks—the prime numbers—which uniquely defines its divisors. By comparing the prime factors of 420 and 630, we identify the common ones and take the lowest power of each to determine the GCD.

Prime Factorization of 420 and 630

Let’s begin by determining the prime factorization of both numbers.

Step 1: Factor 420

Start by dividing 420 by the smallest prime, 2:
420 ÷ 2 = 210
210 ÷ 2 = 105
105 is not divisible by 2, so we try 3:
105 ÷ 3 = 35
35 is not divisible by 3, so try 5:
35 ÷ 5 = 7
7 is prime.

Key Insights

So, the prime factorization of 420 is:
420 = 2² × 3 × 5 × 7

Step 2: Factor 630

Now factor 630:
630 ÷ 2 = 315
315 is not divisible by 2, so divide by 3:
315 ÷ 3 = 105
105 ÷ 3 = 35
35 ÷ 5 = 7
7 is prime.

Thus, the prime factorization of 630 is:
630 = 2 × 3² × 5 × 7

Step 3: Find Common Prime Factors

Now list the primes appearing in both factorizations:

• 2: appears in both, powers are 2 in 420 and 1 in 630 → use 2¹
  
• 3: appears in both, 1 in 420 and 2 in 630 → use 3¹
  
• 5: appears in both → use 5¹
  
• 7: appears in both → use 7¹

Final Thoughts

Lowest powers selected give:
GCD = 2¹ × 3¹ × 5¹ × 7¹ = 2 × 3 × 5 × 7

Step 4: Calculate the GCD

Multiply step by step:
2 × 3 = 6
6 × 5 = 30
30 × 7 = 210

Thus, the greatest common divisor of 420 and 630 is 210.

Why This Method Works

Using prime factorization ensures no divisor is missed and guarantees accuracy. It applies universally, whether working with simple integers or complex numbers in advanced mathematics or computer algorithms. Once prime factorizations are known, deriving the GCD involves identifying shared primes and taking powers — simple and efficient.