A question for set-theorists

1) Let every unit interval after a natural number on the real axis be coloured white with exception of the powers of 2 which are coloured black. Is it possible to shift the black intervals so that the whole real axis becomes black?
2) Let every unit interval after a natural number on the real axis be coloured as above with exception of the intervals after the odd prime numbers which are coloured red. Is it possible to shift the red intervals so that the whole real axis becomes red?
What colour has the real axis after you have solved both tasks?
Regards, WM