111…000…???

Q&ACategory: Mathematics111…000…???
Avik DasAvik Das asked 2 years ago

Let, n be any natural no. Then prove that, it has a nonzero multiple whose representation in (base 10) contains only digits 0 and 1

1 Answers
Avik DasAvik Das answered 2 years ago

We know that, there are only n distinct numbers modulo n. Let us consider the set S={1,11,111,1111,…} whose largest no. contains (n+1) digits. Then we must get atleast 2 no.s in “S” whose modulo n are same. So, if we subtract those two no.s, the new no. will be divisible by n and that no. will contains digits 0 and 1 only.
Hence, proved