Question

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

Short Answer

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Question: Prove that the smallest exponent \(m\) for which \(a^m \equiv 1\bmod p\), where \(p\) is a prime number and \(0<a<p\), is a divisor of \(p-1\). Answer: The smallest exponent \(m\) for which \(a^m \equiv 1\bmod p\) is a divisor of \(p-1\) because applying Fermat's Little Theorem and utilizing a proof by contradiction shows that if \(m\) does not divide \(p-1\), it contradicts the assumption of it being the smallest such exponent. Therefore, it must be true that the smallest exponent \(m\) is a divisor of \(p-1\).

Step by step solution

Recall Fermat's Little Theorem

Fermat's Little Theorem states that if \(p\) is a prime number and \(a\) is an integer not divisible by \(p\) (i.e., \(0<a<p\)), then \(a^{(p-1)} \equiv 1(\bmod p)\). In this problem, we are given that \(p\) is prime and \(0<a<p\), so we can apply Fermat's Little Theorem.

Apply Fermat's Little Theorem

By applying Fermat's Little Theorem, we know that \(a^{(p-1)} \equiv 1 (\bmod p)\). This means that the smallest exponent \(m\) for which \(a^m \equiv 1(\bmod p)\) is less than or equal to \(p-1\). We can denote the smallest such \(m\) as \(m_0\). Now, let's assume that \(m_0 \nmid (p-1)\). This means that \(p-1 = m_0 \cdot k + r\), where \(0 < r < m_0\).

Calculate \(a^{(p-1)}\) using \(m_0\) and \(r\)

Using our assumption, we can express \(p-1\) as \(m_0 \cdot k + r\). We have: $$a^{(p-1)} = a^{(m_0\cdot k + r)} = (a^{m_0})^k \cdot a^r \equiv 1^k\cdot a^r\equiv a^r (\bmod p)$$ This is because \(a^{m_0} \equiv 1(\bmod p)\) by definition of \(m_0\).

Prove that \(r = 0\) by contradiction

Now, we know that \(a^{(p-1)} \equiv a^r (\bmod p)\), and by Fermat's Little Theorem, we also know that \(a^{(p-1)} \equiv 1(\bmod p)\). Therefore, \(a^r \equiv 1 (\bmod p)\). However, this contradicts our initial assumption, because we assumed that \(m_0\) is the smallest exponent for which \(a^m \equiv 1 (\bmod p)\). Since \(0<r<m_0\), we know that \(m_0\) is no longer the smallest such exponent. This contradiction implies that our assumption that \(m_0 \nmid (p-1)\) is false. Therefore, it must be true that \(m_0 \mid (p-1)\).

Conclusion

We have shown that the smallest exponent \(m\) for which \(a^m \equiv 1 (\bmod p)\) must be a divisor of \(p-1\).

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Key Concepts

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

Understanding Prime Numbers

Prime numbers are the building blocks of our numerical system, serving as a foundational element for various concepts in number theory. These numbers are greater than 1 and have no divisors other than 1 and themselves. This means they can't be formed by multiplying two smaller natural numbers together. For example, 2, 3, 5, and 7 are prime numbers, while 4 (which can be expressed as 2x2) is not.

The importance of prime numbers in mathematics stretches beyond just number theory. Their properties make them crucial for algorithms in computer security, such as encryption, and their relative 'unpredictability' poses great challenges as well as opportunities within the discipline. In the context of Fermat's Little Theorem, prime numbers take center stage, as the theorem makes assumptions specifically about prime numbers and their behavior in modular arithmetic.

Demystifying Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers 'wrap around' after reaching a certain value, known as the modulus. Think of it as the arithmetic version of a clock: when you go past 12, you start back at 1. The phrase 'a is congruent to b modulo n', denoted as \(a \equiv b (\bmod\ n)\), means that when a and b are divided by n, they leave the same remainder. This concept is pivotal in many areas of mathematics, including cryptography, computer science, and, as seen in our exercise, number theory. Understanding modular arithmetic is not only key to grasping Fermat’s Little Theorem but is also immensely useful in simplifying complex problems involving divisibility.

The Structure of Mathematical Proof

A mathematical proof is a logical argument that demonstrates the truth of a given statement. Proofs are the backbone of mathematical logic and can take many forms, including direct proof, proof by contradiction, proof by induction, and more. The goal of a proof is to use established truths, definitions, and axioms to show that a new statement necessarily follows. In the step by step problem-solving method we're examining, we use a proof by contradiction, which assumes the opposite of what we want to prove and shows that this assumption leads to a logical inconsistency, thereby proving that the original statement must be true. This method often illuminates hidden relationships and properties within a problem, leading to a deeper understanding of the concept at hand. In Fermat's Little Theorem, proof by contradiction plays a crucial role in validating the relationship between the base, exponent, and prime modulus.

Exponentiation and its Powers

Exponentiation is a mathematical operation that raises a base number to the power of an exponent. In simple terms, the exponent tells us how many times to multiply the base by itself. For instance, \(a^3\) means \(a \times a \times a\). A crucial aspect of exponentiation when working with modular arithmetic, as we see in Fermat's Little Theorem, is that it follows specific rules that aid in simplifying complex expressions. These rules include the power of a product, the product of powers, and the power of a power, all of which are essential tools for effectively working through problems involving large exponents. Additionally, understanding exponentiation is indispensable for computing powers in a modular system, which often produces surprising and counterintuitive results, such as the fact that raising a number to a large exponent can yield a small remainder when divided by a prime number.

Short answer

Question: Prove that the smallest exponent \(m\) for which \(a^m \equiv 1\bmod p\), where \(p\) is a prime number and \(0<a<p\), is a divisor of \(p-1\). Answer: The smallest exponent \(m\) for which \(a^m \equiv 1\bmod p\) is a divisor of \(p-1\) because applying Fermat's Little Theorem and utilizing a proof by contradiction shows that if \(m\) does not divide \(p-1\), it contradicts the assumption of it being the smallest such exponent. Therefore, it must be true that the smallest exponent \(m\) is a divisor of \(p-1\).

Step-by-step solution

Recall Fermat's Little Theorem

Fermat's Little Theorem states that if \(p\) is a prime number and \(a\) is an integer not divisible by \(p\) (i.e., \(0<a<p\)), then \(a^{(p-1)} \equiv 1(\bmod p)\). In this problem, we are given that \(p\) is prime and \(0<a<p\), so we can apply Fermat's Little Theorem.

Apply Fermat's Little Theorem

By applying Fermat's Little Theorem, we know that \(a^{(p-1)} \equiv 1 (\bmod p)\). This means that the smallest exponent \(m\) for which \(a^m \equiv 1(\bmod p)\) is less than or equal to \(p-1\). We can denote the smallest such \(m\) as \(m_0\). Now, let's assume that \(m_0 \nmid (p-1)\). This means that \(p-1 = m_0 \cdot k + r\), where \(0 < r < m_0\).

Calculate \(a^{(p-1)}\) using \(m_0\) and \(r\)

Using our assumption, we can express \(p-1\) as \(m_0 \cdot k + r\). We have: $$a^{(p-1)} = a^{(m_0\cdot k + r)} = (a^{m_0})^k \cdot a^r \equiv 1^k\cdot a^r\equiv a^r (\bmod p)$$ This is because \(a^{m_0} \equiv 1(\bmod p)\) by definition of \(m_0\).

Prove that \(r = 0\) by contradiction

Now, we know that \(a^{(p-1)} \equiv a^r (\bmod p)\), and by Fermat's Little Theorem, we also know that \(a^{(p-1)} \equiv 1(\bmod p)\). Therefore, \(a^r \equiv 1 (\bmod p)\). However, this contradicts our initial assumption, because we assumed that \(m_0\) is the smallest exponent for which \(a^m \equiv 1 (\bmod p)\). Since \(0<r<m_0\), we know that \(m_0\) is no longer the smallest such exponent. This contradiction implies that our assumption that \(m_0 \nmid (p-1)\) is false. Therefore, it must be true that \(m_0 \mid (p-1)\).

Conclusion

We have shown that the smallest exponent \(m\) for which \(a^m \equiv 1 (\bmod p)\) must be a divisor of \(p-1\).

Key concepts

Understanding Prime Numbers

Prime numbers are the building blocks of our numerical system, serving as a foundational element for various concepts in number theory. These numbers are greater than 1 and have no divisors other than 1 and themselves. This means they can't be formed by multiplying two smaller natural numbers together. For example, 2, 3, 5, and 7 are prime numbers, while 4 (which can be expressed as 2x2) is not.

The importance of prime numbers in mathematics stretches beyond just number theory. Their properties make them crucial for algorithms in computer security, such as encryption, and their relative 'unpredictability' poses great challenges as well as opportunities within the discipline. In the context of Fermat's Little Theorem, prime numbers take center stage, as the theorem makes assumptions specifically about prime numbers and their behavior in modular arithmetic.

Demystifying Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers 'wrap around' after reaching a certain value, known as the modulus. Think of it as the arithmetic version of a clock: when you go past 12, you start back at 1. The phrase 'a is congruent to b modulo n', denoted as \(a \equiv b (\bmod\ n)\), means that when a and b are divided by n, they leave the same remainder. This concept is pivotal in many areas of mathematics, including cryptography, computer science, and, as seen in our exercise, number theory. Understanding modular arithmetic is not only key to grasping Fermat’s Little Theorem but is also immensely useful in simplifying complex problems involving divisibility.

The Structure of Mathematical Proof

A mathematical proof is a logical argument that demonstrates the truth of a given statement. Proofs are the backbone of mathematical logic and can take many forms, including direct proof, proof by contradiction, proof by induction, and more. The goal of a proof is to use established truths, definitions, and axioms to show that a new statement necessarily follows. In the step by step problem-solving method we're examining, we use a proof by contradiction, which assumes the opposite of what we want to prove and shows that this assumption leads to a logical inconsistency, thereby proving that the original statement must be true. This method often illuminates hidden relationships and properties within a problem, leading to a deeper understanding of the concept at hand. In Fermat's Little Theorem, proof by contradiction plays a crucial role in validating the relationship between the base, exponent, and prime modulus.

Exponentiation and its Powers

Exponentiation is a mathematical operation that raises a base number to the power of an exponent. In simple terms, the exponent tells us how many times to multiply the base by itself. For instance, \(a^3\) means \(a \times a \times a\). A crucial aspect of exponentiation when working with modular arithmetic, as we see in Fermat's Little Theorem, is that it follows specific rules that aid in simplifying complex expressions. These rules include the power of a product, the product of powers, and the power of a power, all of which are essential tools for effectively working through problems involving large exponents. Additionally, understanding exponentiation is indispensable for computing powers in a modular system, which often produces surprising and counterintuitive results, such as the fact that raising a number to a large exponent can yield a small remainder when divided by a prime number.