Prove that, given any positive integer n, some multiple of it must be of the form 99...900...0

Question

Prove that, given any positive integer n, some multiple of it must be of the form 99...900...0 Give me a hand, please.

http://math.stackexchange.com/questions/164986/smallest-multiple-whose-digits-are-only-ones-and-zeros — lab bhattacharjee, Oct 14 '14 at 18:09

score 4 · Answer 1 · answered Oct 14 '14 at 18:09

4

Hint: use Dirichlet pigeonhole principle to show that some two numbers of the form 999...99 (all nines) give the same remainder when divided by n.

answered Oct 14 '14 at 18:09

Wojowu

26,600

score 1 · Answer 2 · answered Oct 14 '14 at 18:10

1

If $n\in\mathbb N$, then $n=2^k5^\ell z$, where $z$ not divisible by $2$ or $5$. But $z$ divides $10^{z-1}-1$, and hence $n$ divides $10^{\max{k,\ell}}(10^{z-1}-1)$.

answered Oct 14 '14 at 18:10

Yiorgos S. Smyrlis

83,933

Prove that, given any positive integer n, some multiple of it must be of the form 99...900...0

2 Answers2