Greatest Common Factor Of 100 And 75

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Introduction

In mathematics, the greatest common factor (GCF) is the largest positive integer that divides two or more numbers without leaving a remainder. Finding the GCF of two numbers is essential in simplifying fractions, reducing polynomial expressions, and solving equations. In this article, we will discuss how to find the GCF of 100 and 75.

Prime Factorization

One of the most common methods to find the GCF of two numbers is by prime factorization. Prime factorization is the process of breaking down a number into its prime factors, which are the prime numbers that can divide the number without leaving a remainder. To find the prime factorization of 100 and 75, we can use the following steps: - Divide the number by the smallest prime number that can divide it evenly. - Repeat step 1 until the quotient is a prime number. Using this method, we can get the prime factorization of 100 as 2 x 2 x 5 x 5 and the prime factorization of 75 as 3 x 5 x 5.

Common Factors

Once we have found the prime factorization of the two numbers, we can identify the common factors by multiplying the prime factors that are shared by both numbers. In this case, the common factors of 100 and 75 are 5 x 5, which is equal to 25.

Greatest Common Factor

To find the GCF of 100 and 75, we need to identify the largest factor that the two numbers have in common. In this case, the GCF of 100 and 75 is 25, which is the largest factor that the two numbers share.

Euclidean Algorithm

Another method to find the GCF of two numbers is by using the Euclidean algorithm. The Euclidean algorithm is a recursive algorithm that involves finding the remainder when one number is divided by the other. The steps to find the GCF of 100 and 75 using the Euclidean algorithm are as follows: - Divide the larger number by the smaller number. - Find the remainder. - Replace the larger number with the smaller number and the smaller number with the remainder. - Repeat steps 1 to 3 until the remainder is zero. - The GCF is the last nonzero remainder. Using this method, we can get the following calculations: 100 / 75 = 1 remainder 25 75 / 25 = 3 remainder 0 Therefore, the GCF of 100 and 75 is 25.

Conclusion

In conclusion, the greatest common factor of 100 and 75 is 25. We can find the GCF by using either prime factorization or the Euclidean algorithm. Finding the GCF of two numbers is an important skill in mathematics and can be applied in various areas such as simplifying fractions, reducing polynomial expressions, and solving equations.