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➕pre-algebra review

key term - Greatest Common Factor (GCF)

Citation:

Definition

The greatest common factor (GCF) is the largest positive integer that divides two or more integers without a remainder. It is a fundamental concept in number theory and is particularly important in the context of fractions, as it can be used to simplify and compare fractions.

5 Must Know Facts For Your Next Test

  1. The GCF of two or more numbers can be found by first finding the prime factorization of each number, and then selecting the common prime factors with the lowest exponents.
  2. The GCF is important in simplifying fractions, as it can be used to reduce the numerator and denominator to their lowest terms.
  3. Finding the GCF can also be useful in solving problems involving the multiplication and division of fractions, as it can help to identify common factors that can be canceled out.
  4. The GCF of two numbers is always less than or equal to the smaller of the two numbers.
  5. If the GCF of two numbers is 1, then the numbers are said to be relatively prime.

Review Questions

  • Explain how the GCF can be used to simplify fractions.
    • The GCF can be used to simplify fractions by dividing both the numerator and denominator by the GCF of the two numbers. This reduces the fraction to its lowest terms, making it easier to compare and perform operations with. For example, if the fraction is $\frac{24}{36}$, the GCF of 24 and 36 is 12. Dividing both the numerator and denominator by 12 gives $\frac{2}{3}$, which is the simplified form of the original fraction.
  • Describe the relationship between the GCF and the Least Common Multiple (LCM) of two numbers.
    • The GCF and the LCM of two numbers are closely related. The product of the GCF and the LCM of two numbers is equal to the product of the two numbers. This means that if you know the GCF and one of the numbers, you can calculate the other number by dividing the product of the two numbers by the GCF. Conversely, if you know the LCM and one of the numbers, you can calculate the other number by dividing the product of the two numbers by the LCM.
  • Analyze the significance of the GCF in the context of multiplying and dividing fractions.
    • The GCF plays a crucial role in the multiplication and division of fractions. When multiplying fractions, the GCF of the numerators and the GCF of the denominators can be used to simplify the resulting fraction. When dividing fractions, the GCF of the numerator of the dividend and the denominator of the divisor can be used to cancel out common factors, making the division process more efficient. Understanding the GCF and how to apply it in these operations is essential for accurately and efficiently working with fractions.

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