Number Theory

Problem 8. Prove that there exists an integer number whose decimal representation consists entirely of 1'sand which is divisible by 1999.

Solution. We look at the ``pigeons'' numbered $1, 11, 111, ..., \underbrace{111...1}_{2000}$. At least two of these are going to have equal remainders when divided by 1999. Then their difference is representable as $(111...1)\cdot10^k$, and since 10^k is relatively prime with 1999, the first factor should be the number we were seeking for. Problem 9. Prove that there exist two powers of 2 which differ by 1999.