gcf of 45 and 18

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30-03-2025

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Finding the greatest common factor (GCF), also known as the greatest common divisor (GCD), of two numbers is a fundamental concept in mathematics. This article will guide you through several methods to determine the GCF of 45 and 18. We'll explore the prime factorization method and the Euclidean algorithm, providing a clear understanding of each process. By the end, you'll be able to confidently calculate the GCF of any two numbers.

Method 1: Prime Factorization

The prime factorization method involves breaking down each number into its prime factors. The GCF is then found by identifying the common prime factors and multiplying them together.

Step 1: Find the prime factorization of 45.

45 can be factored as 3 x 3 x 5, or 3² x 5.

Step 2: Find the prime factorization of 18.

18 can be factored as 2 x 3 x 3, or 2 x 3².

Step 3: Identify common prime factors.

Both 45 and 18 share two factors of 3.

Step 4: Multiply the common prime factors.

3 x 3 = 9

Therefore, the GCF of 45 and 18 is $\boxed{9}$.

Method 2: The Euclidean Algorithm

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

Step 1: Divide the larger number (45) by the smaller number (18).

45 ÷ 18 = 2 with a remainder of 9.

Step 2: Replace the larger number with the smaller number, and the smaller number with the remainder.

Now we find the GCF of 18 and 9.

Step 3: Repeat the division.

18 ÷ 9 = 2 with a remainder of 0.

Step 4: The last non-zero remainder is the GCF.

The last non-zero remainder was 9.

Therefore, the GCF of 45 and 18 is $\boxed{9}$.

Understanding the Greatest Common Factor

The GCF represents the largest number that divides both 45 and 18 without leaving a remainder. This concept is crucial in simplifying fractions and solving various mathematical problems. For example, simplifying the fraction 45/18 would involve dividing both the numerator and denominator by their GCF (9), resulting in the simplified fraction 5/2.

Applications of the GCF

Finding the GCF has applications beyond simplifying fractions. It's used in:

• Algebra: Factoring polynomials often requires finding the GCF of the terms.
• Geometry: Determining the dimensions of the largest square that can tile a rectangle relies on the GCF of the rectangle's sides.
• Number Theory: The GCF plays a vital role in various number theory concepts, such as modular arithmetic.

Conclusion

Determining the greatest common factor of 45 and 18, whether using prime factorization or the Euclidean algorithm, reveals the answer to be 9. Understanding these methods provides valuable tools for tackling more complex mathematical problems. Remember to choose the method that best suits your needs and the numbers involved. The GCF is a foundational concept with wide-ranging applications across different mathematical areas.