Lesson 3: Greatest Common Factor (GCF) and Applications

Property

The greatest common factor, GCF, of two numbers is that common factor that is the largest. Once we have the prime factorization of two numbers, the GCF is the product of all the prime factors common to both numbers. When we have a prime as a multiple factor of both numbers, the GCF takes the smaller of the multiples.

Examples

• Find the GCF of 48 and 36. Factors of 48: {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}. Factors of 36: {1, 2, 3, 4, 6, 9, 12, 18, 36}. The GCF is 12.
• Find the GCF of 60 and 84 using prime factors. $60 = 2 \cdot 2 \cdot 3 \cdot 5$. $84 = 2 \cdot 2 \cdot 3 \cdot 7$. The common factors are $2 \cdot 2 \cdot 3$, so the GCF is 12.
• Use the GCF to simplify the sum $28 + 42$. Factors of 28: {1, 2, 4, 7, 14, 28}. Factors of 42: {1, 2, 3, 6, 7, 14, 21, 42}. The GCF is 14. Or using prime factors, $28 = 2 \cdot 2 \cdot 7$, $42 = 2 \cdot 3 \cdot 7$, the common factors are $2 \cdot 7$, so the GCF is 14. Then $28 + 42 = 14 \times 2 + 14 \times 3 = 14(2+3) = 14 \times 5 = 70$.

Explanation

The GCF is the biggest number that divides evenly into two or more numbers. It's the king of all the common factors! We use it to simplify fractions and expressions to their simplest forms.

Property

In equal grouping problems, the GCF of two or more quantities represents the maximum number of identical groups that can be formed with no items left over.
Each group will contain $\frac{\text{total items}}{\text{GCF}}$ of each type of item.

Examples