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A short class with Sarah and her students. The other ways are under More.
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Today, we're going to explore linear congruences. Can anyone tell me how a linear equation differs from a linear congruence?
In a linear equation, we usually deal with equals, like ax = b. But in congruences, we use the symbol ≡.
Correct! In a linear equation, we're finding an exact solution, while in a congruence, we're looking for values that give the same remainder. For example, if we have 6x ≡ 4 (mod 10), what does that mean?
It means that when we divide 6x and 4 by 10, they leave the same remainder!
Exactly! You can express it as 6x - 4 is divisible by 10. Can anyone think of a simple solution to this?
If x = 4, then 6(4) = 24, which is congruent to 4 modulo 10.
Well explained! And there can be infinite solutions, right? Like x = 4 + 10k. Now, let’s summarize what linear congruences mean.
Linear congruences involve modular relationships, allowing for infinite solutions defined by base values plus multiples of the modulus.
Next, let’s discuss how to solve linear congruences. One method we can use is the extended Euclidean algorithm. Does anyone know under what condition we can use this method?
It works when GCD(a, N) is 1?
Spot on! So let’s say we have a = 3, N = 11, and we want to solve 3x ≡ 2 (mod 11). What’s our first step?
We need to find the multiplicative inverse of a, which is 3.
Right! If we apply the extended Euclidean algorithm, how can we find this inverse?
Using the algorithm to find integers such that 3y + 11k = 1.
Exactly! Let’s summarize the method for solving linear congruences using the extended Euclidean algorithm.
We compute the multiplicative inverse if GCD(a, N) = 1, and then solve x ≡ b * a^(-1) (mod N).
Now let’s transition to the Chinese Remainder Theorem. How many of you have heard of it?
It’s about solving systems of congruences, right?
Correct! The CRT helps to find a unique solution when the moduli are pairwise coprime. Can someone give me an example of a system of congruences?
Like if x ≡ 2 (mod 3), x ≡ 3 (mod 5), and x ≡ 2 (mod 7).
Excellent! The CRT assures us that there will be one unique solution modulo the product of the moduli. How do we find this solution?
By constructing special linear combinations of the remainders?
Yes! Each combinator relates to a specific modulus. Let’s summarize what we’ve learned about the Chinese Remainder Theorem.
The Chinese Remainder Theorem provides a technique to solve simultaneous linear congruences, yielding a unique solution modulo the product of the moduli.
When numbers align, their remainders stay fine, in modular land, they join hand in hand.
Imagine a baker using 3 different ovens (moduli) to bake cakes, and each oven has different requirements (remainders). The baker must figure out how to satisfy all ovens at once to make the best cake!
For systems of congruences, remember 'C.R.E.A.M.' - Congruences, Remainders, Extended Euclidean, and Multiply.
CMR: Congruences, Moduli, Remainders for remembering CRT conditions.
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Linear Congruence
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Extended Euclidean Algorithm
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Chinese Remainder Theorem
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A relationship of the form ax ≡ b (mod N), where a, b, and N are integers.
A method for finding the GCD of two integers and expressing it as a linear combination of those integers.
A theorem that provides a way to solve a system of simultaneous linear congruences.
An integer b such that a × b ≡ 1 (mod N).
A set of numbers where the GCD of every pair is 1.
This section introduces linear congruences and two methods for solving them: the extended Euclid’s algorithm and the Chinese Remainder Theorem.
In this section, we explore the fundamentals of linear congruences, explaining how they differ from traditional linear equations. Additionally, we discuss two approaches to solving these congruences, particularly highlighting the extended Euclid’s algorithm and the Chinese Remainder Theorem as effective methods.
In this section, we delve into the concept of linear congruences, illustrated first by the familiar linear equation ax = b. Here, we introduce the notion of congruence, which implies the relationship of numbers under modular arithmetic. For example, the congruence relation a x ≡ b (mod N) specifies that the difference ax - b is divisible by N. We then provide an example where a solution to the equation 6x ≡ 4 (mod 10) can yield infinite solutions of the form 4 + 10k and 9 + 10k, for any integer k.
Next, we focus on two pivotal methods to solve linear congruences. The first method utilizes the extended Euclid's algorithm to find the multiplicative inverse, which is possible if the greatest common divisor (GCD) of a and N is 1. If GCD(a, N) ≠ 1, a different approach is needed, leading us into the second core method: the Chinese Remainder Theorem (CRT).
The CRT addresses systems of linear congruences when the moduli are pairwise co-prime. We articulate the theorem's statement, emphasizing that a unique solution exists modulo the product of the individual moduli, allowing us to express solutions in terms of their remainders. The section concludes with a detailed explanation of finding the unique solution leveraging careful construction of special linear combinations and the properties inherent in modular arithmetic.