Greatest Common Factor Of 84 And 105

Finding the Greatest Common Factor (GCF) of 84 and 105

Finding the greatest common factor (GCF), also known as the greatest common divisor (GCD), of two numbers is a fundamental concept in mathematics with applications in various fields, from simplifying fractions to solving algebraic equations. This article will guide you through several methods to determine the GCF of 84 and 105, explaining the process clearly and concisely. Understanding these methods will equip you with the skills to find the GCF of any pair of numbers.

What is the Greatest Common Factor (GCF)?

The greatest common factor (GCF) of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. In simpler terms, it's the biggest number that goes evenly into both numbers. For example, the GCF of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18 without leaving a remainder.

Methods for Finding the GCF of 84 and 105

We'll explore three common methods: listing factors, prime factorization, and the Euclidean algorithm.

1. Listing Factors

This method involves listing all the factors of each number and identifying the largest common factor.

• Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
• Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105

By comparing the two lists, we can see that the common factors are 1, 3, 7, and 21. The largest of these is 21. Therefore, the GCF of 84 and 105 is 21. This method works well for smaller numbers but can become cumbersome with larger numbers.

2. Prime Factorization

This method involves finding the prime factorization of each number and then identifying the common prime factors raised to the lowest power.

• Prime factorization of 84: 2² × 3 × 7
• Prime factorization of 105: 3 × 5 × 7

The common prime factors are 3 and 7. The lowest power of 3 is 3¹ and the lowest power of 7 is 7¹. Multiplying these together gives us 3 × 7 = 21. Thus, the GCF of 84 and 105 is 21. This method is generally more efficient than listing factors for larger numbers.

3. Euclidean Algorithm

The Euclidean algorithm is a highly efficient method for finding the GCF, especially for larger numbers. It involves repeatedly applying the division algorithm until the remainder is 0. The last non-zero remainder is the GCF.

1. Divide the larger number (105) by the smaller number (84): 105 ÷ 84 = 1 with a remainder of 21.
2. Replace the larger number with the smaller number (84) and the smaller number with the remainder (21): 84 ÷ 21 = 4 with a remainder of 0.
3. Since the remainder is 0, the GCF is the last non-zero remainder, which is 21.

The Euclidean algorithm provides a systematic and efficient way to find the GCF, regardless of the size of the numbers.

Conclusion

We have demonstrated three different methods to find the greatest common factor of 84 and 105. All three methods lead to the same answer: 21. Choosing the best method depends on the numbers involved and your personal preference. For smaller numbers, listing factors might suffice. For larger numbers, prime factorization or the Euclidean algorithm are more efficient and less prone to errors. Understanding these methods provides a solid foundation for tackling more complex mathematical problems involving factors and divisors.