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Chapter 5: Problem 21

Establish the congruence $$ 2222^{5555}+5555^{2222} \equiv 0(\bmod 7) $$

Short Answer

Expert verified
The expression is not congruent to 0 modulo 7.

Step by step solution

01

Fermat's Little Theorem

According to Fermat's Little Theorem, if \( p \) is a prime and \( a \) is an integer not divisible by \( p \), then \( a^{p-1} \equiv 1 \pmod{p} \). Here, we are working with \( p = 7 \). Therefore, for any integer \( a \) that is not divisible by 7, \( a^6 \equiv 1 \pmod{7} \).
02

Simplifying Exponents with Fermat's Theorem

We need to compute \( 2222^{5555} + 5555^{2222} \pmod{7} \). Simplify the odd exponents using Fermat's Theorem. We have \( 5555 \equiv 1 \pmod{6} \) and \( 2222 \equiv 2 \pmod{6} \) since \( 5555 \mod 6 = 1 \) and \( 2222 \mod 6 = 2 \). Hence, \( 2222^{5555} \equiv 2222^1 \equiv 2222 \pmod{7} \) and \( 5555^{2222} \equiv 5555^2 \pmod{7} \).
03

Calculate 2222 Modulo 7

We compute \( 2222 \pmod{7} \). Dividing 2222 by 7 gives a remainder of 3, so \( 2222 \equiv 3 \pmod{7} \).
04

Calculate 5555虏 Modulo 7

Compute \( 5555 \pmod{7} \). Dividing 5555 by 7, we get a remainder of 4, so \( 5555 \equiv 4 \pmod{7} \). Therefore, \( 5555^2 \equiv 4^2 \equiv 16 \equiv 2 \pmod{7} \).
05

Establishing the Congruence

Combine the results from Steps 3 and 4: \( 2222^{5555} + 5555^{2222} \equiv 3 + 2 \equiv 5 \pmod{7} \). Therefore, the result is not congruent to 0, thereby not satisfying \( \equiv 0 \pmod{7} \).

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Key Concepts

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

Modular Arithmetic
Modular arithmetic is a mathematical system that focuses on the remainder when dividing one number by another. It's like a clock face that loops around after reaching a certain point. For example, on a 12-hour clock, after 12 o'clock, we start back at 1 o'clock.
In mathematics, we express this idea using the modulus operator. For example, "7 mod 3" equals 1, because 7 divided by 3 leaves a remainder of 1.
  • It helps simplify problems by removing repetitive patterns.
  • Particularly useful in computational algorithms and cryptography.
  • Operations like addition, subtraction, and multiplication work the same, but always check the remainder.
In our specific exercise, we're dealing with terms like \(2222^{5555} \mod 7\), breaking down complex exponents into simple remainders.
Congruence
Congruence is a way to express equality in modular arithmetic. We say two numbers are congruent modulo a number if they have the same remainder when divided by that number.
For example, 8 is congruent to 2 modulo 6, since both leave a remainder of 2 when divided by 6.
  • It is written as \(a \equiv b \pmod{n}\), meaning both \(a\) and \(b\) leave the same remainder when divided by \(n\).
  • It's an equivalence relation, meaning it has properties like reflexivity, symmetry, and transitivity.
  • Crucial for solving equations involving large numbers by reducing them to manageable forms.
In solving the exercise, we use congruence to simplify \(2222 \equiv 3 \pmod{7}\) by determining the remainder, making calculations easy.
Number Theory
Number theory explores the properties and relationships of integers. It's a fundamental branch of mathematics that deals with the study of numbers and the intricacies within.
Some basic elements include prime numbers, divisibility rules, and describing integers in various forms.
  • Primes are a central concept, as every number can factored into primes.
  • Includes famous theorems like Fermat's Little Theorem and the Chinese Remainder Theorem.
  • Applications range from computer science to cryptography due to its role in secure keys.
In our solution, Fermat's Little Theorem, a part of number theory, allows us to cut down large exponents, given the modulus is a prime, saving computation effort.

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Most popular questions from this chapter

Factor the number \(2^{11}-1\) by Fermat's factorization method.

Show that \(561 \mid 2^{561}-2\) and \(561 \mid 3^{561}-3 .\) It is an unanswered question whether there exist infinitely many composite numbers \(n\) with the property that \(n \mid 2^{n}-2\) and \(n \mid 3^{n}-3 .\)

Use Fermat's theorem to verify that 17 divides \(11^{104}+1\).

Establish the statements below: (a) If the number \(M_{p}=2^{p}-1\) is composite, where \(p\) is a prime, then \(M_{p}\) is a pseudoprime. (b) Every composite number \(F_{n}=2^{2^{*}}+1\) is a pseudoprime \((n=0,1,2, \ldots)\).

Use Fermat's method to factor each of the following numbers: (a) 2279 (b) 10541 (c) 340663
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