# Integer mod proof

## Homework Statement

Prove that either $$[a]_{n}$$$$\cap$$$$_{n}$$=empty set or $$[a]_{n}$$=$$_{n}$$.

## The Attempt at a Solution

I want to assume there is an element x in $$[a]_{n}$$$$\cap$$$$_{n}$$ and show this implies $$[a]_{n}$$=$$_{n}$$.
This tells me x is in $$[a]_{n}$$ and $$_{n}$$.
That's where I get stuck.

Last edited:

## Answers and Replies

sorry, I meant to show this implies [a]=.

Office_Shredder
Staff Emeritus
Gold Member
If x=a mod n then n|x-a. x=b mod n means n|x-b.

If x|x-a and n|x-b.... what can you say about a-b?

x-a=x-b
x-x=a-b
0=a-b
b=a

Office_Shredder
Staff Emeritus