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Chapter 1: Problem 2

Find the greatest common divisor of 100 and 104 .

Short Answer

Expert verified
The greatest common divisor of 100 and 104 is 4.

Step by step solution

01

Identification of Factors

Start by listing the factors of each number. The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, 100. The factors of 104 are 1, 2, 4, 8, 13, 26, 52, 104.
02

Common Factors

Next, identify the common factors from the lists of factors. The common factors of 100 and 104 are 1, 2, and 4.
03

Greatest Common Factor

From the common factors identified, choose the greatest one. The greatest common factor of 100 and 104 is 4.

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Key Concepts

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

Factors
To understand the greatest common divisor (GCD), we first need to grasp the concept of factors. Factors of a number are integers that can exactly divide the number without leaving a remainder.

For example, let's take 100. Its factors are: 1, 2, 4, 5, 10, 20, 25, 50, and 100. This means all these numbers can divide 100 completely. Similarly, 104 has factors like 1, 2, 4, 8, 13, 26, 52, and 104.

These lists of factors will help us later in identifying the common factors shared between the numbers.
Common Factors
Once we have the factors of each number, the next step is to identify the common factors. Common factors are the numbers that appear in the factor lists of both numbers.

In our previous example, we listed the factors for 100 and 104. By comparing these lists, we find the common factors: 1, 2, and 4.

These common factors are useful to determine the greatest common divisor, as they help us understand which numbers can divide both original numbers entirely.
Number Theory
Number theory is the branch of mathematics that deals with the properties and relationships of numbers. It's essential for understanding more complex problems involving factors and divisors.

In number theory, the concept of divisors plays a crucial role. It's the foundation upon which we build more complex ideas, such as finding the GCD. Understanding prime factorization, multiples, and the relationships between different numbers can help in solving these problems.

In our example, knowledge of number theory helps us systematically list factors and identify common ones, leading to finding the GCD.
GCD Calculation
The greatest common divisor (GCD) of two numbers is the largest number that can divide both without leaving a remainder. To calculate the GCD, we use the factors and common factors.

Let's revisit our example: we first found the factors of 100 and 104, then identified the common factors: 1, 2, and 4. Among these, the greatest one is 4. Thus, the GCD of 100 and 104 is 4.

There are other methods to calculate the GCD, such as the Euclidean algorithm. However, understanding the basic method of finding common factors is fundamental and can be easily applied.

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