Answer to Question #312677 in Discrete Mathematics for anj

Question

Prove that there is a positive multiple of 3333 which is entirely made of  0s and 1s. (For example: 110000011; note that we don’t need to find  the number. We just need to prove that there exists such a number) 

Answer 1

$N=111111111111$ is multiple of 1111 since

$N=1111\cdot 100010001$

Also N is divisible by 3 since the sum of its digits is 12, which is divisible by 3.

Since 1111 and 3 are mutually prime, from divisibility of N by 1111 and by 3 we have that N is multiple of 3333.