Solved Ring Theory Question: Integer Relatively Prime to n

[ehrenfest]

[SOLVED] ring theory question

Homework Statement

My book says that "if a is an integer relatively prime to n, then the coset a +nZ of nZ containing contains an integer b < n and relatively prime to n."

If 0< a < n, this is obvious. If a > n or 0 > a, I do not see why that statement is true.

Homework Equations

The Attempt at a Solution

 

[ehrenfest]

Actually, I do see why it is true. By the division algorithm, we have

a=qn+r, where 0 <= r < n

If gcd(n,r)=d>1, then d must divide a.

So, b=r.