Congruence Class Proofs: Tips and Examples

[Joe20]

Hi, I have tried the question as attached. I am not sure if I am correct. You help is greatly appreciated. Thanks in advance!

 

[GJA]

Hi Alexis87,

I like the idea to examine each of the cases. A few were left out - e.g., $[a]==[2]$ - and I don't know if that was intentional on your part or not. It may be a little neater to note that the result of squaring in $\mathbb{Z}_{3}$ can only be $[0]$ or $[1]$, in which case there are only three possibilities to check. Your counterexample looks good.

 

[HallsofIvy]

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.

 

[Joe20]

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.

Hi HallsofIvy,

may I ask what's the purpose of showing this expression:
=3(3n3+3m3+2an+2bm)+a2+b2=0 ?

Is there any thing we can interpret from this equation?

So for "So we must have a and b less than 3" does it mean that because it is Z subscript 3?

 

[HOI]

The purpose is to realize that $3(3n^3+ 3m^3+ 2an+ 2bn)$ is a multiple of 3 and so can be ignored "modulo 3". That is why I can say "We must have a and b positive integers less than 3 such that $a^2+ b^2= 0$.