Abstract Algebra: Congruence and The Division Algorithm 1
I've come to bother you again. This is a homework problem, so please, offer only hints if you can, especially since I think this problem should be easy for me (it has the stench of something that is straight forward).
Prove that if is a positive integer, and , then there is an integer such that
Things that may come in handy:
As the title suggests, I have a strong gut feeling that I'm supposed to use the Division Algorithm here. But I can't seem to make it fit together nicely.
The Division Algorithm: If with , then there exists unique integers and such that
What I've Tried:
Okay, so I decided to try and make this work by the division algorithm.
Now, means that for
By the Division Algorithm, if with , then there are unique integers and such that
...and I'm stuck there...
I was thinking of choosing , and . so the integer I am looking for would be
but I don't think that proves anything, nor am I sure that I can actually choose them like that...
Thanks guys (and gals -- dedicated to JaneBennet)