Problem 5 Show that if \(p\) and \(q\) are... [FREE SOLUTION]

Chapter 6: Problem 5

Show that if \(p\) and \(q\) are distinct primes, then \(p^{q-1}+q^{p-1} \equiv 1(\bmod p q)\).

Short Answer

Expert verified

By applying Fermat's Little theorem, we obtain the congruences \(q^{p-1} \equiv 1 (\bmod p)\) and \(p^{q-1} \equiv 1 (\bmod q)\). Adding these congruences, we get \(q^{p-1} + p^{q-1} \equiv 2 (\bmod p, q)\). Since p and q are distinct primes, their least common multiple is pq. Hence, \(q^{p-1} + p^{q-1} \equiv 2 (\bmod pq)\). Subtracting 1 from both sides, we arrive at the desired result: \(p^{q-1} + q^{p-1} \equiv 1 (\bmod pq)\).

Step by step solution

Fermat's Little Theorem for p and q

We will use Fermat's Little theorem for the prime numbers p and q. It states that for any prime p and any integer a not divisible by p, we have \(a^{p-1} \equiv 1 (\bmod p)\). Since p and q are distinct prime numbers, they are not divisible by each other, so we can apply Fermat's Little theorem to both p and q.

Applying Fermat's Little theorem to p and q

Using Fermat's Little theorem, we get the following congruences: \(q^{p-1} \equiv 1 (\bmod p)\) and \(p^{q-1} \equiv 1 (\bmod q)\).

Adding the congruences

Now, we will add the two congruences obtained in Step 2: \((q^{p-1} \equiv 1 (\bmod p)) + (p^{q-1} \equiv 1 (\bmod q))\) The left side adds up to \(q^{p-1} + p^{q-1}\) and the right side adds up to \(1 + 1\). Thus, we have: \(q^{p-1} + p^{q-1} \equiv 1 + 1 (\bmod p, q)\) \(q^{p-1} + p^{q-1} \equiv 2 (\bmod p, q)\)

Check for the congruence modulo pq

We have found that \(q^{p-1} + p^{q-1} \equiv 2\hspace{5pt}(\bmod p, q)\). We now need to verify if the congruence is true modulo pq. If a number is congruent to 2 modulo p and modulo q, the number must be congruent to 2 modulo the least common multiple of p and q, which is their product pq since they are coprime (distinct prime numbers). Therefore, we have: \(q^{p-1} + p^{q-1} \equiv 2 (\bmod pq)\) Subtracting 1 from both sides, we get the desired result: \(p^{q-1} + q^{p-1} \equiv 1 (\bmodpq)\) Hence, we have successfully shown that if p and q are distinct prime numbers, then \(p^{q-1} + q^{p-1} \equiv 1 (\bmod pq)\).

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Key Concepts

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

modular arithmetic

Modular arithmetic works like the clock we use every day. Instead of going beyond a certain number, we start over from the beginning. For example, on a clock, after 12 comes 1. In modular arithmetic, this process is defined by the "modulus." The modulus is the number at which counting resets.
Consider you have 17 cookies and you want to divide them among 5 children. After each child gets 3 cookies, you are left with 2 cookies. We say 17 is "congruent to" 2 mod 5, written as:

17 鈮� 2 (mod 5)

The key concept is the remainder; modular arithmetic focuses on this part. Similar calculations can easily be handled with other numbers and larger moduli. Therefore, it simplifies many calculations, especially involving division, by keeping everything in small whole numbers.

congruences

In the world of mathematics, a congruence is an expression of equality under a certain modulus. When two numbers leave the same remainder when divided by a given number, they are said to be congruent to each other.
The concept is particularly useful in solving problems related to divisibility and helps simplify large numbers by analyzing them in segments. For example, both 14 and 24 are congruent to 4 modulo 5 since:

14 divided by 5 gives a remainder of 4.
24 divided by 5 gives a remainder of 4.

Congruences are governed by a few crucial rules that make calculations more manageable:

If two numbers are congruent modulo a number, adding, subtracting, or multiplying will yield a result that is also congruent under the modulus.

Therefore, congruences allow us to operate on numbers without leaving the same modulus environment, ensuring simplicity and consistency in mathematical expressions and operations.

prime numbers

Prime numbers are the basic building blocks of mathematics. A prime number is any whole number greater than 1 that has only two divisible factors, namely, 1 and itself. For instance, the number 7 is prime because only 1 and 7 divide it without a remainder.
Prime numbers are crucial in various branches of mathematics. They form the backbone of number theory and are central to concepts like encryption in computer science. Their unique properties make them indispensable for Secure Systems and Cryptography.
Here are some important points about prime numbers:

All even numbers greater than 2 are not prime since they are divisible by 2.
The number 1 is not considered a prime number.
Primes cannot be formed by multiplying other numbers together, which adds to their uniqueness.

Being able to identify and use prime numbers is important, especially in areas involving factors and divisibility, enabling deeper understanding and problem solving in broader contexts.

91影视

Show that if \(p\) and \(q\) are distinct primes, then \(p^{q-1}+q^{p-1} \equiv 1(\bmod p q)\).

Short Answer

Step by step solution

Fermat's Little Theorem for p and q

Applying Fermat's Little theorem to p and q

Adding the congruences

Check for the congruence modulo pq

Key Concepts

modular arithmetic

congruences

prime numbers

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