Question

Prove that if \(p\) is a prime number and \(0

Short Answer

Expert verified

The largest number that can divide both a prime number \(p\) and any positive integer \(h\) less than \(p\) is 1. Hence, the greatest common divisor (gcd) of \(p\) and \(h\) is always 1.

Step by step solution

Understand the properties of prime numbers

Prime numbers are numbers that have only two factors: 1 and themselves. Therefore, any number less than a prime number, like \(h\) in this case, cannot be a factor of the prime number \(p\). This is because \(h\) is strictly less than \(p\), and by definition, \(p\) has no factors other than 1 and itself.

Consider the possible gcd values

The Greatest Common Divisor (gcd) of two numbers is the largest number that divides both of them without leaving a remainder. Since from step 1 we know \(h\) cannot be a factor of \(p\), the gcd of \(p\) and \(h\) cannot be \(h\) itself or any value greater than 1. The gcd also cannot be 0, as zero doesn't divide anything.

Conclude the proof

By the principle of elimination, the only possible gcd left is 1. Therefore, we can establish that gcd\((p, h)=1\).

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Key Concepts

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

Greatest Common Divisor (GCD)

To understand the exercise, it's essential to grasp the concept of the Greatest Common Divisor (GCD). The GCD of two numbers is the largest number that divides both of them without leaving a remainder.
This means if two numbers share a divisor, the GCD is this common factor.

• If the GCD is 1, the numbers are considered coprime, or relatively prime, meaning they share no factors other than 1.
• For example, the GCD of 8 and 12 is 4, which is the largest number that can divide both evenly.
• However, the GCD of 9 and 28 is 1, indicating no shared factors except for 1.

This concept is crucial in number theory, especially when analyzing relationships between numbers. It simplifies expressions and aids in solving mathematical problems like this one.

Prime Numbers

Prime numbers are a key element in number theory. They are numbers greater than 1 that have no divisors other than 1 and themselves.
These numbers form the building blocks of other numbers, as every integer greater than 1 can be represented as a product of prime numbers (known as prime factorization).

• A prime number cannot be divided evenly by any numbers other than 1 and itself.
• Examples of prime numbers include 2, 3, 5, 7, 11, and 13.
• The number 2 is unique as it is the only even prime number.

Understanding primes is foundational in proving properties related to numbers since they represent the simplest form of divisors. In our exercise, knowing that a prime number has no other divisors except 1 and itself helps establish that any smaller number cannot divide it. This property aids in demonstrating that if you take any number less than a prime, their GCD with the prime is guaranteed to be 1.

Mathematical Proofs

Mathematical proofs are logical arguments demonstrating the truth of a given statement.
These are essential for establishing the validity of mathematical assertions.

• Proofs rely on logical reasoning and often use properties of numbers, such as the definitions and properties of prime numbers or GCDs.
• They involve a series of conclusions that start from known truths, usually arising from axioms or previously established results.
• For instance, the exercise utilizes the definition of a prime number to argue that the GCD of a prime number with any lesser number is 1.

The conclusion in the step-by-step solution hinges on these logical deductions. By applying the properties of prime numbers and GCDs step by step, we arrive at a solid proof. These reasoning structures are fundamental for mathematically proving why certain attributes or qualities exist in numbers.

Short answer

The largest number that can divide both a prime number \(p\) and any positive integer \(h\) less than \(p\) is 1. Hence, the greatest common divisor (gcd) of \(p\) and \(h\) is always 1.

Step-by-step solution

Understand the properties of prime numbers

Prime numbers are numbers that have only two factors: 1 and themselves. Therefore, any number less than a prime number, like \(h\) in this case, cannot be a factor of the prime number \(p\). This is because \(h\) is strictly less than \(p\), and by definition, \(p\) has no factors other than 1 and itself.

Consider the possible gcd values

The Greatest Common Divisor (gcd) of two numbers is the largest number that divides both of them without leaving a remainder. Since from step 1 we know \(h\) cannot be a factor of \(p\), the gcd of \(p\) and \(h\) cannot be \(h\) itself or any value greater than 1. The gcd also cannot be 0, as zero doesn't divide anything.

Conclude the proof

By the principle of elimination, the only possible gcd left is 1. Therefore, we can establish that gcd\((p, h)=1\).

Key concepts

Greatest Common Divisor (GCD)

To understand the exercise, it's essential to grasp the concept of the Greatest Common Divisor (GCD). The GCD of two numbers is the largest number that divides both of them without leaving a remainder.
This means if two numbers share a divisor, the GCD is this common factor.

• If the GCD is 1, the numbers are considered coprime, or relatively prime, meaning they share no factors other than 1.
• For example, the GCD of 8 and 12 is 4, which is the largest number that can divide both evenly.
• However, the GCD of 9 and 28 is 1, indicating no shared factors except for 1.

This concept is crucial in number theory, especially when analyzing relationships between numbers. It simplifies expressions and aids in solving mathematical problems like this one.

Prime Numbers

Prime numbers are a key element in number theory. They are numbers greater than 1 that have no divisors other than 1 and themselves.
These numbers form the building blocks of other numbers, as every integer greater than 1 can be represented as a product of prime numbers (known as prime factorization).

• A prime number cannot be divided evenly by any numbers other than 1 and itself.
• Examples of prime numbers include 2, 3, 5, 7, 11, and 13.
• The number 2 is unique as it is the only even prime number.

Understanding primes is foundational in proving properties related to numbers since they represent the simplest form of divisors. In our exercise, knowing that a prime number has no other divisors except 1 and itself helps establish that any smaller number cannot divide it. This property aids in demonstrating that if you take any number less than a prime, their GCD with the prime is guaranteed to be 1.

Mathematical Proofs

Mathematical proofs are logical arguments demonstrating the truth of a given statement.
These are essential for establishing the validity of mathematical assertions.

• Proofs rely on logical reasoning and often use properties of numbers, such as the definitions and properties of prime numbers or GCDs.
• They involve a series of conclusions that start from known truths, usually arising from axioms or previously established results.
• For instance, the exercise utilizes the definition of a prime number to argue that the GCD of a prime number with any lesser number is 1.

The conclusion in the step-by-step solution hinges on these logical deductions. By applying the properties of prime numbers and GCDs step by step, we arrive at a solid proof. These reasoning structures are fundamental for mathematically proving why certain attributes or qualities exist in numbers.