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Chapter 4: Problem 8

Find the GCD of the given numbers. $$10,40$$

Short Answer

Expert verified
The GCD of 10 and 40 is 10.

Step by step solution

01

Identify the Factors

First, identify all the factors of each number. For 10, the factors are 1, 2, 5, and 10. For 40, the factors are 1, 2, 4, 5, 8, 10, 20, and 40.
02

Find Common Factors

Identify the common factors from the lists obtained in Step 1. The common factors of 10 and 40 are 1, 2, 5, and 10.
03

Choose the Greatest Common Factor

Out of the common factors 1, 2, 5, and 10, the greatest one is 10. Thus, the greatest common divisor (GCD) of 10 and 40 is 10.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factors
In mathematics, factors are integers that you can multiply together to get another number. They serve as building blocks to make up larger numbers. To find factors, you start by dividing the number by smaller integers to see if they divide evenly without leaving a remainder.
  • For example, factors of 10 are 1, 2, 5, and 10 because these numbers can divide 10 without any remainder. Specifically, \( 1 \times 10 = 10 \) and \( 2 \times 5 = 10 \).
  • Similarly, for 40, factors include 1, 2, 4, 5, 8, 10, 20, and 40 because 40 can be broken down into pairs like \( 1 \times 40 \), \( 2 \times 20 \), \( 4 \times 10 \), and \( 5 \times 8 \).
Identifying factors is the first step when solving problems involving the Greatest Common Divisor (GCD). By listing the factors, you prepare the ground for finding what these numbers have in common.
Common Factors
Once you have lists of factors for more than one number, the next step is to identify common factors. Common factors are numbers that appear in the factor lists of each number being analyzed.
  • For our example with the numbers 10 and 40, the factors of 10 are 1, 2, 5, and 10.
  • For 40, the factors are 1, 2, 4, 5, 8, 10, 20, and 40.
The common factors between these two sets are 1, 2, 5, and 10, since these numbers can divide both 10 and 40 without leaving a remainder.
Identifying these shared numbers is crucial because it sets up the next step: finding the greatest common factor, the most powerful tool in simplifying ratios, solving for unknowns, and revealing simpler mathematical relationships.
Greatest Common Factor
The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD), is the largest number that is a factor of two or more numbers. It plays a key role in simplifying fractions, rationalizing denominators, and enhancing number comprehension.
  • From the common factors of 10 and 40, which are 1, 2, 5, and 10, the greatest one is 10.
  • This means that 10 is the largest factor that divides both numbers evenly.
Finding the GCD can not only simplify math problems but also provide insights into the relationships between numbers and their divisors, making it a fundamental concept in number theory and its applications.

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