SMO 2012 Junior Round 1 Question 8

Question 8

For a real number $x$, let $\lfloor x\rfloor$ be the largest integer at most $x$, and let $\{x\}=x-\lfloor x\rfloor$. Real numbers $a,b$, with $b\ne0$, satisfy
$$a=b\left\lfloor\frac ab\right\rfloor-b\left\{\frac ab\right\}.$$
Which statement is incorrect?

If b is an integer, then a is an integer.
If a is a nonzero integer, then b is an integer.
If b is rational, then a is rational.
If a is a nonzero rational number, then b is rational.
If b is an even number, then a is an even number.