In some book about elementary number theory I found a theorem that when two integers $a$ and $b$ are both divisible by the same common factor $f$, then their sum $a+b$ is also divisible by the same factor. In short:
$f|a \land f|b \implies f|(a+b)$
There's a corollary annexed to the above theorem which says that when some factor $f$ divides a whole (the sum) and one of its parts ($a$, for example), then it also divides the other part ($b$, correspondingly).
My question is: Is it enough information to draw that inverse conclusion?
The book says so, and I know it is true, because I tested it for different cases and I have proven it geometrically (one cannot split a rectangle into two smaller rectangles without them having a common edge=factor). But I'm a bit suspicious about the validity of this corollary being there in the book without any further proof to show that the converse is also true, because I know that implications cannot be simply reversed. Are my suspicions right?
Best Answer
If $f$ divides $a$ then $f$ divides $-a$. If $f$ also divides $a+b$ then $f$ divides $(a+b)+(-a)$ by the theorem you stated, and $(a+b)+(-a) = b$.
You are right that you can't simply reverse the arrows, but it does follow quite simply (at least, it certainly isn't many lines of work, obviously how obvious something is is subjective).