Solution: To find the greatest common factor (GCF) of 48 and 60, factor both numbers. - Dyverse

Understanding the Context

The GCF of two numbers is the largest positive integer that divides both numbers without leaving a remainder. To determine this, factoring each number into its prime components reveals common factors, making the GCF easy to calculate.

Step 1: Factor Both Numbers into Primes

Factoring 48:
We start by dividing 48 by the smallest prime numbers:

• 48 is even, so divide by 2:
  48 ÷ 2 = 24
  
• 24 is also even:
  24 ÷ 2 = 12
  
• Continue dividing by 2:
  12 ÷ 2 = 6
  
• And again:
  6 ÷ 2 = 3
  
• Finally, 3 is a prime number.

So, the prime factorization of 48 is:
48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3¹

Key Insights

Factoring 60:
Divide 60 by the smallest primes:

• 60 is even:
  60 ÷ 2 = 30
  
• 30 is even:
  30 ÷ 2 = 15
  
• 15 is divisible by 3:
  15 ÷ 3 = 5
  
• 5 is prime.

So, the prime factorization of 60 is:
60 = 2 × 2 × 3 × 5 = 2² × 3¹ × 5¹

Step 2: Identify Common Prime Factors

Now, compare the prime factorizations:

• 48 = 2⁴ × 3¹
  
• 60 = 2² × 3¹ × 5¹

The common prime bases are 2 and 3. For each common prime, use the lowest exponent present in both factorizations:

• For 2: minimum exponent is 2 (from 60)
  
• For 3: minimum exponent is 1 (common to both)

Final Thoughts

Step 3: Compute the GCF

Multiply the common primes raised to their lowest exponents:
GCF = 2² × 3¹ = 4 × 3 = 12

Conclusion

The greatest common factor of 48 and 60 is 12. Factoring both numbers reveals their shared prime basis, making it simple to determine the GCF. This method—prime factorization and identifying common factors—is reliable, efficient, and a key strategy for mastering GCF across all grades of mathematics.

Whether you're solving equations, simplifying fractions, or preparing for advanced math concepts, understanding how to find the GCF through factoring is an invaluable skill. Practice with other numbers to strengthen your factoring skills, and soon, identifying the GCF will feel effortless!