Just as a precursor, I'm not a mathematics major (though I've had some experience with abstract algebra and number theory), so I have absolutely no idea how to approach this problem. Hints, not solutions, would be preferred. ^^
Prove that every positive integer can be multiplied by a power of 2 such that the decimal expansion of the product has at least as many 8's as it has 4's.
My first thought was, "Hey, this question seems to have a big loophole! If the decimal expansion has zero 4's, it's allowed to have zero 8's!"
But I couldn't really find a way to go down that route either. I also don't know if multiplying by ${2^0}$ is allowed.
Any ideas?