Chapter 1: Problem 3
Find the greatest common divisor of -30 and 95.
Short Answer
Expert verified
Answer: The greatest common divisor of -30 and 95 is 5.
Step by step solution
01
1. Verify that the numbers are integers
Verify that the numbers -30 and 95 are both integers.
02
2. Make both numbers positive
To ensure a non-negative result, make both numbers positive by taking their absolute values: |-30| = 30 and |95| = 95.
03
3. Now the problem is to find the GCD of 30 and 95
To find the GCD, we can use the Euclidean Algorithm, which is an efficient way of finding the GCD of two numbers. The algorithm is as follows:
1. If either of the numbers is zero, the GCD is the other number.
2. If both numbers are non-zero, divide the larger number by the smaller number, and replace the larger number with the remainder.
3. Repeat step 2 until the remainder is zero.
04
4. Apply the Euclidean Algorithm
Now, perform the steps from the Euclidean Algorithm for the numbers 30 and 95.
1. Remainder of 95 divided by 30: \(95 \div 30 = 3 \, remainder 5\)
2. Remainder of 30 divided by 5: \(30 \div 5 = 6 \, remainder 0\)
The remainder is now zero, so the GCD is the last non-zero remainder we obtained, which was 5.
05
5. State the GCD
The greatest common divisor of -30 and 95 is 5.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Euclidean Algorithm
Understanding the Euclidean Algorithm can significantly simplify the task of finding the greatest common divisor (GCD) of two numbers. Originating from ancient Greek mathematics, the algorithm is based on a simple yet powerful process that involves repeatedly dividing and obtaining remainders until you reach zero.
For example, to find the GCD of 30 and 95, we first divide the larger number (95) by the smaller number (30) to get a remainder of 5 after three complete divisions (since 95 divided by 30 equals 3 with a remainder of 5). Next, we take the original smaller number (30) and divide it by the remainder (5). This time, the division is exact; 30 divided by 5 equals 6 with a remainder of 0. When we reach a remainder of zero, the GCD is the last non-zero remainder, which in this case is 5.
This process is not only simple but also universally applicable regardless of the size of the integers involved, making it a foundational tool in number theory.
For example, to find the GCD of 30 and 95, we first divide the larger number (95) by the smaller number (30) to get a remainder of 5 after three complete divisions (since 95 divided by 30 equals 3 with a remainder of 5). Next, we take the original smaller number (30) and divide it by the remainder (5). This time, the division is exact; 30 divided by 5 equals 6 with a remainder of 0. When we reach a remainder of zero, the GCD is the last non-zero remainder, which in this case is 5.
This process is not only simple but also universally applicable regardless of the size of the integers involved, making it a foundational tool in number theory.
Absolute Value
The concept of absolute value is crucial when dealing with the greatest common divisor (GCD), especially if the numbers involved are negative. Absolute value refers to the distance of a number from zero on the number line, without regard to direction. In mathematical terms, it's denoted with vertical bars and is always non-negative.
To illustrate, consider the numbers -30 and 95. Their absolute values are |-30| = 30 and |95| = 95. When finding the GCD, we take the absolute values to ensure we work with non-negative integers, since the GCD is inherently a positive quantity or zero. The absolute value simplifies many mathematical procedures and helps maintain consistency in the real-world interpretation of quantities like distance, which, like the GCD, cannot be negative.
To illustrate, consider the numbers -30 and 95. Their absolute values are |-30| = 30 and |95| = 95. When finding the GCD, we take the absolute values to ensure we work with non-negative integers, since the GCD is inherently a positive quantity or zero. The absolute value simplifies many mathematical procedures and helps maintain consistency in the real-world interpretation of quantities like distance, which, like the GCD, cannot be negative.
Integers
Integers are the set of whole numbers that include all positive numbers, negative numbers, and zero. They do not contain fractions or decimals and form a foundation for many mathematical concepts including the greatest common divisor (GCD). The set of integers is usually denoted by the symbol \( \mathbb{Z} \), coming from the German word 'Zahlen' which means 'numbers'.
In our example, both -30 and 95 are integers. The significance of confirming that both numbers are integers lies in the fact that the Euclidean Algorithm specifically requires two non-zero integers to function properly. This is because the algorithm involves division and finding remainents, operations that are well-defined for integers and align with the natural concept of numbers as counts or measures.
In our example, both -30 and 95 are integers. The significance of confirming that both numbers are integers lies in the fact that the Euclidean Algorithm specifically requires two non-zero integers to function properly. This is because the algorithm involves division and finding remainents, operations that are well-defined for integers and align with the natural concept of numbers as counts or measures.
Number Theory
Number theory is a branch of mathematics concerned with the properties and relationships of numbers, particularly the integers. It's often referred to as 'higher arithmetic' due to its deep connections with the simplest operations such as addition, subtraction, multiplication, and division.
The significance of the Euclidean Algorithm within number theory is immense. Finding the GCD of two integers is just one of its many applications within the field. Number theory delves into various topics such as prime numbers, divisibility, modular arithmetic, and Diophantine equations. GCD also plays a role in more advanced concepts like cryptography, where it's used to secure digital communications. Thus, the study of number theory not only enhances our understanding of numbers but also has practical implications in the modern world.
The significance of the Euclidean Algorithm within number theory is immense. Finding the GCD of two integers is just one of its many applications within the field. Number theory delves into various topics such as prime numbers, divisibility, modular arithmetic, and Diophantine equations. GCD also plays a role in more advanced concepts like cryptography, where it's used to secure digital communications. Thus, the study of number theory not only enhances our understanding of numbers but also has practical implications in the modern world.
