key term - Greatest Common Factor

Definition

The greatest common factor (GCF) is the largest positive integer that divides each of the given integers without a remainder. It is a fundamental concept in mathematics that is particularly relevant in the context of fractions, properties of real numbers, and factoring by grouping.

5 Must Know Facts For Your Next Test

1. The GCF is used to simplify fractions by dividing both the numerator and denominator by the GCF.
2. 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.
3. The GCF is a crucial concept in the properties of real numbers, as it helps to understand the relationships between different sets of numbers, such as integers, rational numbers, and irrational numbers.
4. Factoring by grouping is a technique that utilizes the GCF to factor polynomials more efficiently.
5. The GCF can be used to solve real-world problems, such as finding the largest size of a tile that can be used to cover a rectangular floor without leaving any gaps.

Review Questions

• Explain how the greatest common factor (GCF) is used to simplify fractions.
  • The GCF is used to simplify fractions by dividing both the numerator and denominator by the GCF of the two numbers. This reduces the fraction to its simplest form, which is important in mathematical operations involving fractions. 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 fraction.
• Describe the relationship between the GCF and the prime factorization of numbers.
  • 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. This is because the GCF is the largest positive integer that divides each of the given integers without a remainder, and the prime factorization of a number reveals its fundamental building blocks. By identifying the common prime factors and their lowest exponents, you can determine the GCF of the numbers.
• Explain how the concept of the GCF is used in the process of factoring by grouping.
  • Factoring by grouping is a technique that utilizes the GCF to factor polynomials more efficiently. The process involves grouping the terms of the polynomial based on their common factors, and then finding the GCF of the grouped terms. This GCF becomes the common factor that can be factored out, leaving a simpler expression to work with. By identifying and factoring out the GCF, the polynomial can be broken down into smaller, more manageable parts, making the overall factorization process more straightforward.

Related terms

Factor: A factor is a number or expression that divides another number or expression without a remainder.

Prime Factorization:

Prime factorization is the process of expressing a number as a product of its prime factors.

Least Common Multiple (LCM):

The least common multiple (LCM) is the smallest positive integer that is divisible by two or more given integers.