GCF: Greatest Common Factor

Learning Objectives Define factors and common factors of two or more integers. Determine the prime factorization of a given integer. Calculate the Greatest Common Factor (GCF) of two or more integers using the listing factors method. Calculate the Greatest Common Factor (GCF) of two or more integers using the prime factorization method. Apply the concept of GCF to simplify fractions to their lowest terms. Solve real-world problems involving the Greatest Common Factor. Identify when two numbers are relatively prime. Ever wonder how to share things equally among friends or simplify complex fractions? 🤔 Understanding the Greatest Common Factor (GCF) is your secret weapon! In this lesson, you'll learn how to find the largest number that divides into two or more numbers...

TermDefinitionExample FactorA number that divides another number exactly, leaving no remainder.The factors of 12 are 1, 2, 3, 4, 6, 12. Common FactorA factor that two or more numbers share.For 12 and 18, the common factors are 1, 2, 3, 6. Greatest Common Factor (GCF)The largest of the common factors of two or more numbers.For 12 and 18, the GCF is 6. Prime NumberA whole number greater than 1 that has exactly two distinct positive divisors: 1 and itself.2, 3, 5, 7, 11 are prime numbers. Composite NumberA whole number greater than 1 that has more than two distinct positive divisors.4 (factors: 1, 2, 4) and 6 (factors: 1, 2, 3, 6) are composite numbers. Prime FactorizationExpressing a composite number as a product of its prime factors.The prime factorization of 12 is $2 \times 2 \times 3$ or...

Listing Factors Method for GCF To find the GCF of two or more numbers, list all factors for each number, identify the common factors, and then select the largest one. This method is effective for smaller numbers where listing factors is manageable and provides a direct visual understanding of common factors. Prime Factorization Method for GCF To find the GCF of two or more numbers, first find the prime factorization of each number. Then, multiply the common prime factors, raised to the lowest power they appear in any of the factorizations. If $A = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k}$ and $B = p_1^{b_1} p_2^{b_2} \dots p_k^{b_k}$, then $\text{GCF}(A, B) = p_1^{\min(a_1, b_1)} p_2^{\min(a_2, b_2)} \dots p_k^{\min(a_k, b_k)}$. This method is generally more efficient for larger...