Question

Define in the language of arithmetic: (a) \(x\) and \(y\) are relatively prime; (b) \(x\) is the smallest prime greater than \(y\); (c) \(x\) is the greatest number with \(2 x

Short answer

(a) \(\text{gcd}(x, y) = 1\); (b) smallest prime \(x\) with \(x > y\); (c) largest \(x\) with \(2x < y\).

Step-by-step solution

Understanding 'Relatively Prime'

Two integers, \(x\) and \(y\), are relatively prime if their greatest common divisor (GCD) is 1. In arithmetic terms, \(\text{gcd}(x, y) = 1\).

Defining 'Smallest Prime Greater Than'

The smallest prime greater than an integer \(y\) is the smallest integer \(x\) such that \(x\) is a prime number and \(x > y\). This requires checking integers sequentially greater than \(y\) until a prime is found.

Understanding the Condition '2x < y'

Here, \(x\) is defined as the greatest number for which \(2x < y\). This means \(x\) must satisfy the inequality \(2x < y\). Solve for \(x\) by rearranging to \(x < \frac{y}{2}\). Find the greatest integer \(x\) that meets this condition.

Construct Arithmetic Language Definition for (a)

For part (a), "\(x\) and \(y\) are relatively prime" can be defined in arithmetic as \(\text{gcd}(x, y) = 1\). This indicates that the two numbers share no common divisors other than 1.

Construct Arithmetic Language Definition for (b)

For part (b), "\(x\) is the smallest prime greater than \(y\)" is formally described by finding the smallest integer \(x\) such that \(x > y\) and \(x\) is a prime number.

Construct Arithmetic Language Definition for (c)

For part (c), "\(x\) is the greatest number with \(2x < y\)" can be defined by finding the largest integer \(x\) such that the inequality \(2x < y\) holds true.

Key concepts

Relatively Prime Numbers

When two numbers are called relatively prime, it means they don't share any prime factors except for the number 1. In more formal mathematical language, the greatest common divisor, or GCD, of these numbers is 1. This is a fascinating concept because even though the numbers themselves can be quite large, as long as their GCD is 1, they are considered relatively prime. Some characteristics of relatively prime numbers include:

• They do not have any common divisors except 1.
• Examples include the numbers 8 and 15. Their common divisors are just 1, which means they are relatively prime.
• Being relatively prime is not determined by the individual nature of each number, but rather by their relationship to each other.

This is important when dealing with problems involving fractions, as simplifying a fraction is possible by dividing both its numerator and denominator by their GCD.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor, is the highest number that divides exactly into two or more numbers. Finding the GCD is an essential skill in arithmetic and helps in simplifying fractions or in finding least common multiples. Here's a simple way to find the GCD:

• List the factors of each number.
• Identify the largest factor that appears in each list.

For example, consider the numbers 12 and 15. The factors of 12 are 1, 2, 3, 4, 6, and 12, while the factors of 15 are 1, 3, 5, and 15. The largest factor common to both lists is 3, so the GCD is 3. Another method for finding the GCD is using the Euclidean algorithm, which involves dividing numbers and using their remainders to iteratively find the GCD.

Prime Number Identification

Prime numbers are the building blocks of all natural numbers. A prime number is an integer greater than 1 that has no divisors other than 1 and itself. To identify if a number is prime, you should try dividing it by all integers up to its square root. If none of these divisions result in an integer, the number is prime. Key things to remember about prime numbers include:

• The number 2 is the smallest and only even prime number.
• If a number has divisors other than 1 and itself, it is not prime.
• Examples of prime numbers are 2, 3, 5, 7, 11, and 13.

In many arithmetic problems, identifying prime numbers aids in factorization and in discerning patterns or properties of numbers.

Inequality Solutions

Solving inequalities is a crucial skill in mathematics that involves finding the set of values that satisfy a given inequality. In the context of the problem above, we consider the scenario "2x < y". The inequality indicates that we are searching for the greatest integer value of x such that when it is doubled, it remains less than y. To solve this:

• Rearrange the inequality as x < y/2.
• Find the largest integer less than y/2.

For example, if y = 10, then x must be less than 5 because 10/2 = 5. Thus, the integers 0, 1, 2, 3, and 4 all satisfy the inequality, with 4 being the greatest. Solving inequalities helps in various mathematical contexts, including optimization problems and modeling real-world situations.