Greatest Common Factor Of 45 And 36: Explanation And Solution

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Introduction

In mathematics, the greatest common factor (GCF) is the largest number that divides two or more numbers without leaving a remainder. Finding the GCF of two numbers is important in simplifying fractions, finding the least common multiple (LCM), and solving certain types of equations. In this article, we will discuss the GCF of 45 and 36 and how to find it.

Factors of 45 and 36

To find the GCF of 45 and 36, we first need to list the factors of each number. Factors are the numbers that can be multiplied together to get the original number. The factors of 45 are 1, 3, 5, 9, 15, and 45. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Common Factors

After listing the factors of both numbers, we need to identify the factors that they have in common. These are called the common factors. The common factors of 45 and 36 are 1, 3, 9.

Greatest Common Factor

The largest number that is a factor of both 45 and 36 is the GCF. In this case, the GCF of 45 and 36 is 9. This is because 9 is the largest number that both 45 and 36 can be divided by without leaving a remainder.

How to Find the GCF

There are several methods for finding the GCF of two numbers. One method is to list the factors of both numbers as we did earlier, then identify the common factors and choose the largest one. Another method is to use prime factorization.

Prime Factorization

Prime factorization is the process of breaking down a number into its prime factors. Prime factors are the prime numbers that can be multiplied together to get the original number. For example, the prime factorization of 45 is 3 x 3 x 5, and the prime factorization of 36 is 2 x 2 x 3 x 3.

Using Prime Factorization to Find the GCF

To find the GCF using prime factorization, we need to identify the prime factors that both numbers have in common and multiply them together. In this case, both 45 and 36 have a factor of 3, so we can multiply 3 by 3 to get 9. Therefore, the GCF of 45 and 36 is 9.

Conclusion

Finding the GCF of two numbers is an important skill in mathematics. It allows us to simplify fractions, find the LCM, and solve certain types of equations. To find the GCF of 45 and 36, we listed the factors of both numbers, identified the common factors, and chose the largest one. We also used prime factorization to find the GCF. In both cases, we found that the GCF of 45 and 36 is 9.