Lesson 5: Greatest Common Factor

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

To find the prime factorization of a composite number using the tree method:
Step 1. Find any factor pair of the given number, and use these numbers to create two branches.
Step 2. If a factor is prime, that branch is complete. Circle the prime.
Step 3. If a factor is not prime, write it as the product of a factor pair and continue the process.
Step 4. Write the composite number as the product of all the circled primes.

Examples

• To factor 36, start with $6 \cdot 6$. Break the first 6 into $2 \cdot 3$ and circle both primes. Break the second 6 into $2 \cdot 3$ and circle both primes. The factorization is $2^2 \cdot 3^2$, or $2 \cdot 2 \cdot 3 \cdot 3$.
• For 28, branch into $4 \cdot 7$. Circle the prime 7. Branch 4 into $2 \cdot 2$ and circle both 2s. The result is $2^2 \cdot 7$, or $2 \cdot 2 \cdot 7$.
• To factor 60, you can start with $6 \cdot 10$. Branch 6 into $2 \cdot 3$ and 10 into $2 \cdot 5$. All are prime, so the factorization is $2^2 \cdot 3 \cdot 5$, or $2 \cdot 2 \cdot 3 \cdot 5$.

Explanation

This visual method lets you branch out from a number, splitting it into pairs of factors. You keep branching until every path ends in a prime number, which you circle like a bud on the tree.