- #1
HallsofIvy said:I would consider it simpler to write the first number as [a]= 3n+ a and the second as = 3m+ b where a and b are one of 0, 1, or 2. Then $[a]^2+ ^2= (3n+a)^2+ (3m+ b)^2= 9n^2+ 6an+ a^2+ 9m^2+ 6bm+ b^2= 3(3n^3+3m^3+ 2an+ 2bm)+ a^2+ b^2= 0$.
So we must have a and b less than 3 and $a^2+ b^2= 0$. From that, a= 0, b= 0 so that [a]= = 0.
A congruence class is a set of all numbers that have the same remainder when divided by a given modulus. For example, in the congruence class of 4 mod 7, all numbers that have a remainder of 4 when divided by 7 belong to this class (e.g. 11, 18, 25, etc.).
A congruence class proof is a method of proving that two numbers are congruent (have the same remainder when divided by a given modulus) by showing that they belong to the same congruence class. This is often done using algebraic manipulations and the properties of congruence.
Some tips for writing a congruence class proof include: clearly stating the given numbers and modulus, using the properties of congruence (such as addition, subtraction, and multiplication) to manipulate the numbers, and clearly showing each step of the proof.
Sure! Let's prove that 15^2 is congruent to 1 mod 7. First, we can rewrite 15^2 as (14 + 1)^2. Using the distributive property, this becomes 14^2 + 2(14)(1) + 1. Next, we can simplify 14^2 to 196, which is congruent to 0 mod 7. Therefore, the entire expression becomes 0 + 2(14)(1) + 1, which is congruent to 1 mod 7. This shows that 15^2 is indeed congruent to 1 mod 7.
Congruence class proofs are used in various fields of mathematics, such as number theory and abstract algebra. They can also be applied in computer science, particularly in cryptography and coding theory. Additionally, congruence classes have practical applications in fields such as engineering and physics, where they are used in solving equations and modeling systems.