KöMaL 2008 B4070

The positive integers $a$ and $b$, written in decimal system, can be obtained from each other by rearranging their digits. Prove that $a$) the sum of the digits of 2$a$ equals that of 2$b$; $b$) if both $a$ and $b$ are even, then the sum of the digits of $\frac{a}{2}$ equals that of $\frac{b}{2}$.