Divisibility of a sum

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?