10.7 - Construction of the Solution x
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Introduction to Linear Congruences
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Today, we will explore linear congruences, for example, ax ≡ b (mod N). This means we are looking for values of x that give the same remainder when divided by N.
So if we have 6x ≡ 4 (mod 10), how would we find x?
Good question! We can start looking for values of x and check which ones satisfy this condition. Can you try x = 4?
If x = 4, that gives us 24, which is indeed congruent to 4 mod 10!
Exactly! Now can you think of another value that might work?
What about x = 9? That gives us 54, which is also congruent to 4 mod 10!
Perfect! Remember, linear congruences can have many solutions expressed in a general form.
Extended Euclid's Algorithm
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Now let's discuss how to solve a linear congruence using the Extended Euclid’s algorithm. What does it require?
It requires that the gcd(a, N) is equal to 1, right?
That’s correct! If gcd(a, N) = 1, the multiplicative inverse exists. Who can remind us how to apply this method?
We multiply both sides of the congruence by the multiplicative inverse of a?
Exactly! That gives us a solution expressed as x ≡ b*a⁻¹ (mod N).
So does that mean we can just keep adding multiples of N to find more solutions?
Correct! You can always express the infinitely many solutions based on that.
Chinese Remainder Theorem
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Now, let's dive into the Chinese Remainder Theorem. What do we mean by solving a system of linear congruences?
It means we have multiple congruences to satisfy simultaneously?
Exactly! For example, consider x ≡ 2 (mod 3), x ≡ 3 (mod 5), and x ≡ 2 (mod 7). What can we infer?
We need to find an x that satisfies all those conditions!
Right! The CRT assures us of a unique solution modulo the product of the moduli, given they are pairwise coprime.
So once we find one solution, we can find the others by adding multiples of that product?
Yes! You're grasping this well. Let's now proceed to how we can find this special linear combination, which is critical for the CRT.
Constructing Solutions with CRT
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In CRT, we will express the capital modulus as the product of all moduli except the current one. Why do you think this is crucial?
Because it ensures that the summand contributions vanish for all other moduli?
Exactly! Then, each summand's computation modulo the current modulus gives us the remainder we desire.
What if that solution isn't within the range 0 to M-1?
Great question! We can keep adding or subtracting M until we find an appropriate number falling within that range.
So, it's all about manipulating our found solution to fit the conditions?
Absolutely! Let’s pause here and summarize what we’ve learned about linear congruences and the methods to solve them.
Introduction & Overview
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Quick Overview
Standard
In this section, students learn about linear congruences and how to solve them using two different methods. The first method involves utilizing the Extended Euclid’s algorithm to find the multiplicative inverse, and the second method is the more comprehensive Chinese Remainder Theorem which unravels the unique solutions to systems of linear congruences.
Detailed
Detailed Summary of Construction of the Solution x
Linear Congruences
In discrete mathematics, linear congruences are equations of the form ax ≡ b (mod N). Here, our goal is to find the value of x that satisfies this equivalence, meaning that x and b leave the same remainder when divided by N. For example, to solve the congruence 6x ≡ 4 (mod 10), we can find solutions like x = 4 and x = 9, as they satisfy the modular condition. It's important to note that linear congruences can have infinitely many solutions, expressed in the form x = 4 + 10k and x = 9 + 10k, where k can be any integer.
Method 1: Extended Euclid’s Algorithm
The first method for solving linear congruences is through the Extended Euclid’s algorithm, which helps find the multiplicative inverse of a under modulus N. This inverse exists only when the gcd(a, N) = 1. Once the inverse a⁻¹ is computed, we can multiply both sides of the congruence by it to solve for x. Therefore, we express the solution as x ≡ b*a⁻¹ (mod N) + kN, where k is an integer. This method requires that a and N are coprime.
Method 2: Chinese Remainder Theorem (CRT)
The second method to solve linear congruences pertains to the Chinese Remainder Theorem, which is especially useful when dealing with multiple congruence equations. It states that if we have a system of congruences with pairwise coprime moduli, there exists a unique solution modulo the product of the moduli. For example, if we have a system:
- x ≡ 2 (mod 3)
- x ≡ 3 (mod 5)
- x ≡ 2 (mod 7),
once again we can express the solution with some larger modulus, and the CRT guarantees a method to find at least one x satisfying all these congruences simultaneously.
The proof involves constructing a special linear combination of given remainders, utilizing properties of the moduli, and ensuring that these constructions yield valid solutions. Ultimately, this section guides students through both solving individual linear congruences and systems thereof, illustrating the power of modular arithmetic in problem-solving.
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Constructing the Solution
Chapter 1 of 1
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Chapter Content
So now, let us see how exactly we can find at least one solution... if this x that you have obtained as per the Chinese Remainder Theorem satisfies the condition that is it is in the range 0 to M-1, then well and good else, you find out an appropriate multiple or you select an appropriate value of l...
Detailed Explanation
In this portion, the construction of the solution is discussed. It involves finding a special linear combination of the remainders given by the linear congruences and ensuring that it lies within a specified range. The value of x derived using the CRT is shown to be a valid solution, but it may need adjustment to fit within the interval from 0 to M-1. This involves either keeping x as it is if it fits the range or adjusting it by subtracting multiples of M until it lies within the required limits.
Examples & Analogies
Think of a student preparing for a multi-subject exam where each subject has a specific requirement (like study time). By using the CRT, the student can find a study schedule (x) that meets all subject requirements. If the initial schedule overlaps with a previous commitment, instead of scrapping it entirely, the student can adjust their timetable incrementally (subtracting from their schedule) to find available time slots that still satisfy all study conditions.
Key Concepts
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Linear Congruence: An equation of the form ax ≡ b (mod N).
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Extended Euclid's Algorithm: A method to find the multiplicative inverse when gcd(a, N) = 1.
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Chinese Remainder Theorem: A theorem to solve systems of linear congruences with pairwise coprime moduli.
Examples & Applications
Solving the linear congruence 6x ≡ 4 (mod 10) gives solutions x = 4 and x = 9.
Using CRT, from the system x ≡ 2 (mod 3), x ≡ 3 (mod 5), and x ≡ 2 (mod 7), we can find a unique solution modulo 105.
Memory Aids
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Rhymes
To solve with CRT, remember the blend, pairwise primes on which we depend.
Stories
Imagine a farmer with different crops in three fields, needing each field's yield to match sizes 2, 3, 2 when harvested. The CRT helps him find the perfect numbers! Each field is unique, and they work together harmoniously.
Memory Tools
For CRT, remember VCP: Verify coprimes, Construct combinations, and Produce solutions.
Acronyms
C.R.T. - Coprime Roots Together! Helps remember that roots come from coprime steps.
Flash Cards
Glossary
- Linear Congruence
An equation of the form ax ≡ b (mod N), where a, b, and N are integers and x is the unknown.
- Extended Euclid's Algorithm
An algorithm used to find the greatest common divisor of integers and to compute the multiplicative inverse when it exists.
- Chinese Remainder Theorem (CRT)
A theorem that provides a way to solve systems of linear congruences with pairwise coprime moduli, guaranteeing a unique solution modulo their product.
- Multiplicative Inverse
An integer b such that ab ≡ 1 (mod N), only existing if gcd(a, N) = 1.
- Coprime
Two integers are coprime if their greatest common divisor is 1.
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