How do you find the greatest common factor (GCF)?

How do you find the greatest common factor (GCF)?

• A. Determine the largest factor of each number
• B. Add the prime factors
• C. Multiply the common prime factors after factoring
• D. Identify the prime factorization of each integer and multiply the common prime factors

Answer: (D)

The key idea is to use prime factorization and the overlap of prime factors. To find the greatest common factor, factor each number into primes, then look for primes that appear in both factorizations. For each shared prime, use the smaller exponent from the two factorizations, and multiply those primes together. That product is the largest number that divides both original numbers. For example, 48 and 180 factor as 48 = 2^4 · 3 and 180 = 2^2 · 3^2 · 5. The common primes are 2 and 3. Take the smaller exponents: 2^2 and 3^1, and multiply them to get 4 · 3 = 12. This is the greatest factor that divides both numbers. Why not other approaches? Simply taking the largest factor of each number isn’t guaranteed to be common to both. Adding prime factors doesn’t produce a factor, and multiplying common primes without correctly aligning their exponents can give an incorrect result. The precise method—factoring each number and multiplying the common primes with their smallest exponents—gives the correct greatest common factor.

The key idea is to use prime factorization and the overlap of prime factors. To find the greatest common factor, factor each number into primes, then look for primes that appear in both factorizations. For each shared prime, use the smaller exponent from the two factorizations, and multiply those primes together. That product is the largest number that divides both original numbers.

For example, 48 and 180 factor as 48 = 2^4 · 3 and 180 = 2^2 · 3^2 · 5. The common primes are 2 and 3. Take the smaller exponents: 2^2 and 3^1, and multiply them to get 4 · 3 = 12. This is the greatest factor that divides both numbers.

Why not other approaches? Simply taking the largest factor of each number isn’t guaranteed to be common to both. Adding prime factors doesn’t produce a factor, and multiplying common primes without correctly aligning their exponents can give an incorrect result. The precise method—factoring each number and multiplying the common primes with their smallest exponents—gives the correct greatest common factor.