# Thread: natural number divisible by 2004

1. ## natural number divisible by 2004

Hi all;

Prove that there's a natural number divisible by 2004 consisting only of zeros and sevens.

Help appreciated. Thank you.

2. Originally Posted by disclaimer
Hi all;

Prove that there's a natural number divisible by 2004 consisting only of zeros and sevens.
Let $a_1 = 7$, $a_2 = 77$, $a_3 = 777$, .... $a_{2005} = 77 ... 7$.

By pigeonhole principle there are two numbers with same remainder. Thus there are $i>j$ with $a_i - a_j$ divisible by $2004$. And this number only consists of only sevens and zeros.