Hi all;
Prove that there's a natural number divisible by 2004 consisting only of zeros and sevens.
Help appreciated. Thank you.
Let $\displaystyle a_1 = 7$, $\displaystyle a_2 = 77$, $\displaystyle a_3 = 777$, .... $\displaystyle a_{2005} = 77 ... 7$.
By pigeonhole principle there are two numbers with same remainder. Thus there are $\displaystyle i>j$ with $\displaystyle a_i - a_j$ divisible by $\displaystyle 2004$. And this number only consists of only sevens and zeros.