- #1
Leo Liu
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Could someone explain why ##[a][x_0]=[c]\iff ax_0\equiv c\, (mod\, m)##?
My instructor said it came from the definition of congruence class. But I am not convinced.
[Congruence class] Proof of modular arithmetic theorem
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- Thread starter Leo Liu
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- Arithmetic Class Proof Theorem
In summary, a congruence class is a set of integers that are equivalent or congruent to each other in terms of a certain modulus. The modular arithmetic theorem, also known as the division theorem, states that for any integer a and positive integer n, there exists a unique pair of integers q and r such that a = qn + r, where 0 ≤ r < n. This theorem is often used in proofs to show that two integers are congruent to each other, and can also be extended to non-integer numbers. The modular arithmetic theorem has many real-world applications, including cryptography, computer science, and engineering. It is used to solve problems involving remainders and can also be applied to systems with periodic behavior, such
Could someone explain why ##[a][x_0]=[c]\iff ax_0\equiv c\, (mod\, m)##?
My instructor said it came from the definition of congruence class. But I am not convinced.
- #2
ergospherical
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Recall that ##[x] = [y]## if ##x \equiv y \ (\mathrm{mod \ m})##. Then it's just a case of using ##[a][x_0] = [ax_0]##.
- #3
Leo Liu
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Wonderful. I didn't know this form of definition but after seeing its justification (transitivity and symmetry) I am convinced.ergospherical said:
1. What is the Congruence Class Proof of modular arithmetic theorem?
The Congruence Class Proof of modular arithmetic theorem is a mathematical proof that states that if two integers have the same remainder when divided by a positive integer, then they are considered congruent modulo that integer.
2. Why is the Congruence Class Proof of modular arithmetic theorem important?
The Congruence Class Proof of modular arithmetic theorem is important because it allows us to solve problems involving remainders and modular arithmetic in a systematic and efficient way. It also has applications in computer science, cryptography, and other fields.
3. How is the Congruence Class Proof of modular arithmetic theorem used in real life?
The Congruence Class Proof of modular arithmetic theorem is used in real life in a variety of ways. For example, it is used in computer programming to optimize algorithms and in cryptography to ensure secure communication. It is also used in fields such as engineering, finance, and physics to solve problems involving remainders and modular arithmetic.
4. What are some examples of problems that can be solved using the Congruence Class Proof of modular arithmetic theorem?
Some examples of problems that can be solved using the Congruence Class Proof of modular arithmetic theorem include finding the last digit of a large number, determining the day of the week for a given date, and solving linear congruences.
5. Are there any limitations to the Congruence Class Proof of modular arithmetic theorem?
Yes, there are some limitations to the Congruence Class Proof of modular arithmetic theorem. It only applies to integers and cannot be used for non-integer numbers. Additionally, it is only applicable to remainders when divided by a positive integer, not negative integers. It also does not work for all types of modular arithmetic, such as non-linear congruences.
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