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Introduction to Linear Congruences
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Welcome everyone! Today we're diving into linear congruences. Does anyone know how a linear equation translates into the world of modular arithmetic?
I think it relates to solving equations where we're looking for remainders when divided by a number?
Exactly! Linear congruences are like linear equations, but we're interested in remainders instead of exact values. If we have \( ax \equiv b \mod N \), we need to find \( x \) such that both \( x \) and \( b \) give the same remainder when divided by \( N \).
So, can you show us an example of that?
Certainly! For instance, if \( 6x \equiv 4 \mod 10 \), can anyone suggest a solution?
I think \( x = 4 \) works because \( 6 imes 4 = 24 \) which is congruent to 4 modulo 10.
Great! And remember, there's not just one solution; we can express all solutions in the form of \( 4 + 10k \). This highlights the infinitely many solutions concept.
To summarize, linear congruences provide infinitely many solutions, unlike standard linear equations!
Using Extended Euclid’s Algorithm
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Now, let’s discuss how we can solve linear congruences using the Extended Euclid's Algorithm. What condition do we need about the numbers involved?
Is it about the GCD? The GCD of \( a \) and \( N \) must be 1, right?
Exactly! If \( ext{gcd}(a, N) = 1 \), we can find the multiplicative inverse of \( a \) modulo \( N \).
And how do we find that inverse?
We can use the Extended Euclidean Algorithm. If we have our congruence as \( ax \equiv b \mod N \), after finding \( a^{-1} \), we multiply both sides by this inverse to get \( x \equiv b imes a^{-1} ext{ mod } N \).
So, after finding \( a^{-1} \), how do we express our solution?
Correct! The final result is \( x ext{ mod } N = (b imes a^{-1}) ext{ mod } N \) giving us a solution.
To recap, we solve linear congruences when GCD is 1 using the Extended Euclidean Algorithm to find multiplicative inverses!
Chinese Remainder Theorem (CRT)
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Now let's shift gears to the Chinese Remainder Theorem, or CRT. What sets CRT apart from what we've learned so far?
Isn't it about solving multiple linear congruences at once?
Exactly! CRT provides a way to find an unknown \( x \) that satisfies a system of linear congruences, especially when each modulus is pairwise co-prime!
Can you give us an example of how that would work?
Sure! Suppose we have the following system: \( x \equiv 2 ext{ mod } 3 \), \( x \equiv 3 ext{ mod } 5 \), and \( x \equiv 2 ext{ mod } 7 \). How might we approach this?
So, we first establish the equations and ensure that the moduli are co-prime to apply CRT?
That's right! Using CRT, we can express our solution as a linear combination of remainders, leading to unique solutions modulo the product of these moduli.
In summary, CRT efficiently resolves systems of linear congruences when moduli are co-prime, giving unique solutions modulo the product.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, we delve into linear congruences and explore how to solve them using extended Euclid's algorithm, especially when the greatest common divisor (GCD) of the coefficients is one. We also touch on scenarios where the GCD is not one and briefly introduce the Chinese Remainder Theorem for simultaneous congruences.
Detailed
Solving Linear Congruences using Extended Euclid's Algorithm
This section focuses on the topic of linear congruences, which are equations of the form \( ax \equiv b \mod N \). The objective is to find values of \( x \) such that the congruence holds true.
Key Concepts:
- Formulating Linear Congruences: Just as in regular algebra where we solve equations like \( a \cdot x = b \), in modular arithmetic we work with congruences.
- Solutions: Solutions are expressed in terms of a modulus, leading often to infinitely many solutions which can be expressed as \( x = b/a + kN \) for integer \( k \).
- Using Extended Euclid's Algorithm: The primary method discussed for solving these congruences involves using the extended Euclidean algorithm to find the multiplicative inverse of \( a \) modulo \( N \), applicable when \( ext{gcd}(a, N) = 1 \).
- Chinese Remainder Theorem (CRT): In cases where multiple congruences need to be satisfied simultaneously, CRT offers a systematic solution provided the moduli are pairwise coprime. This section will introduce CRT as a way of solving systems of linear congruences, which is significant in various applications in computer science and number theory.
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Introduction to Linear Congruences
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So, now let us see how we can solve linear congruences using extended Euclid’s algorithm that is our method number one. So, you are given a, b and N, goal is to find out this unknown x. Now, as we did for our equation in the linear algebra where we said that divide both sides by a provided a is not 0. The question is can we do something similar in the modular world as well that is can we say divide both sides by a. And divide both sides by a by that I mean multiplying both sides by multiplicative inverse of a. And that is possible only if GCD(a, N) is 1. So, remember, in the earlier; in the last lecture, we proved that the multiplicative inverse modulo N exists only if the number for which you want to find out the inverse is co-prime to your modulus.
Detailed Explanation
In this chunk, the concept of linear congruences is introduced in the context of extended Euclid's algorithm. A linear congruence is similar to a linear equation, but instead of working with real numbers, we deal with integers under a modulus. The goal is to determine an unknown variable x where the linear congruence takes the form 'ax ≡ b (mod N)'. To solve this, we first need to check if the greatest common divisor (GCD) of 'a' and 'N' is 1. If it is, we can find a multiplicative inverse of 'a' in the modular system, which allows us to effectively divide both sides of the congruence by 'a'.
Examples & Analogies
Think of linear congruences like sharing pizzas among friends. If each friend has several pizza slices (represented by integers), but you want to ensure everyone has the same number after sharing (the modulus), you can only divide fairly if the number of slices each friend has and the total number of slices are not entangled (co-prime). If they are, you can easily distribute them evenly (find the inverse).
Using the Extended Euclid's Algorithm
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So, if your GCD(a, N) is 1 then by running the extended Euclid’s algorithm, compute the multiplicative inverse of a namely b, I stress that it is not 1/a in the modular world it is an integer. And now I multiply both the sides of this linear congruence by this a-1. So, I will get this linear congruence and I know that a into a-1 is 1 modulo N and 1 into x modulo N is x. So, I get that x is congruence to b-1 modulo N that means; I can say that the value of x being this plus any multiple of N is a solution for this linear congruence (x = ba-1 mod N + kN).
Detailed Explanation
Once we establish that the GCD(a, N) equals 1, we can find the multiplicative inverse of 'a' using the extended Euclid's algorithm. This gives us an integer 'a⁻¹' such that when multiplied by 'a' gives 1 under modulo N. Multiplying the linear congruence 'ax ≡ b (mod N)' by 'a⁻¹' transforms it to 'x ≡ ba⁻¹ (mod N)'. This means that the value of 'x' can be expressed as 'x =ba⁻¹ + kN', where 'k' can be any integer, thus generating infinitely many solutions of the congruence.
Examples & Analogies
Imagine you've brought various gifts for your friends, and you want to equally distribute them (the congruence). If you have just the right number of gifts (GCD = 1), you can determine how many each friend gets (the inverse). The gifts can come back to you as well (the multiples of N), allowing you to create different combinations of distributions (different values of k).
Handling Non-Coprime Cases
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However, this method will work only if your number a is co-prime to your modulus N. What if the number a is not co-prime to your modulus N, then we have to follow a slightly different approach which is complicated and I am not going to discuss that matter.
Detailed Explanation
This portion outlines a limitation of the method discussed so far. If 'a' and 'N' are not co-prime (i.e., GCD(a, N) is not 1), the existing approach to solve the linear congruence cannot be used directly, and a more complex method would need to be looked into, which goes beyond the introductory explanation in this context. Recognizing when you cannot use the immediate method is important, as it opens up discussions of other techniques or methodologies in modular arithmetic.
Examples & Analogies
Consider trying to share a set of keys among a group. If some of the keys are duplicates and wouldn't fit in some locks (the GCD isn't 1), you can't distribute them equally among your friends in the straightforward way you could if all keys were unique. Instead, you might need to resort to other techniques to find a way to distribute them effectively.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Formulating Linear Congruences: Just as in regular algebra where we solve equations like \( a \cdot x = b \), in modular arithmetic we work with congruences.
-
Solutions: Solutions are expressed in terms of a modulus, leading often to infinitely many solutions which can be expressed as \( x = b/a + kN \) for integer \( k \).
-
Using Extended Euclid's Algorithm: The primary method discussed for solving these congruences involves using the extended Euclidean algorithm to find the multiplicative inverse of \( a \) modulo \( N \), applicable when \( ext{gcd}(a, N) = 1 \).
-
Chinese Remainder Theorem (CRT): In cases where multiple congruences need to be satisfied simultaneously, CRT offers a systematic solution provided the moduli are pairwise coprime. This section will introduce CRT as a way of solving systems of linear congruences, which is significant in various applications in computer science and number theory.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Finding the multiplicative inverse of 3 modulo 11.
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Resolving the system of linear congruences: x ≡ 2 (mod 3), x ≡ 3 (mod 5), and x ≡ 2 (mod 7).
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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When a and N are coprime,
📖 Fascinating Stories
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Imagine Alice and Bob navigating through a maze of lockers, each locker a modulus. They know that if they can find the right keys (the inverses), they can unlock the treasures (solutions) hidden within. This adventure represents solving linear congruences.
🧠 Other Memory Gems
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Remember 'A C for CRT': 'A' for 'Apply', 'C' for 'Coprime'.
🎯 Super Acronyms
INVERSE
- Inverse Needed for Valid Equation in modular Arithmetic.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Linear Congruence
Definition:
An equation of the form \( ax \equiv b \mod N \) representing equivalence of \( ax \) and \( b \) under modulus \( N \).
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Term: Extended Euclid's Algorithm
Definition:
An algorithm to compute the greatest common divisor and the coefficients (multiplicative inverses) of a and b in the context of modular arithmetic.
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Term: Chinese Remainder Theorem (CRT)
Definition:
A theorem that provides conditions under which multiple linear congruences can be solved simultaneously, yielding one unique solution modulo the product of the moduli.
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Term: Multiplicative Inverse
Definition:
An integer \( x \) such that \( ax \equiv 1 \mod N \), existing when \( ext{gcd}(a, N) = 1 \).