The Greatest Common Factor (GCF) is the largest number that divides two or more numbers without leaving a remainder. Finding the GCF of 15 and 35 is a straightforward process. Let’s walk through the steps to solve this problem.
Step 1: Understand the Problem
We need to determine the greatest number that can evenly divide both 15 and 35. To do this, we can use one of several methods: listing factors, using prime factorization, or the Euclidean algorithm.
Step 2: List the Factors
Start by listing all the factors of each number:
- Factors of 15: 1, 3, 5, 15
- Factors of 35: 1, 5, 7, 35
Next, identify the common factors:
- Common Factors: 1, 5
The greatest common factor is the largest of these: 5.
Step 3: Use Prime Factorization
Another way to find the GCF is through prime factorization:
- Prime factorization of 15: 3 × 5
- Prime factorization of 35: 5 × 7
Identify the common prime factors:
- Both numbers have a common factor of 5.
Thus, the GCF is 5.
Step 4: Verify Using the Euclidean Algorithm
The Euclidean algorithm provides an efficient way to find the GCF:
- Divide the larger number by the smaller number and find the remainder:
- remainder
- Replace the larger number with the smaller number and repeat:
- remainder
- When the remainder is 0, the divisor (5) is the GCF.
The Greatest Common Factor (GCF) of 15 and 35 is 5. Knowing how to find the GCF is a valuable skill for simplifying fractions, solving problems in number theory, and many practical applications in everyday life.