For $n$ houses arranged in a circle, a man goes clockwise skipping $i$ houses every time. For what value of $n$ does he visit each house?

Question

A man wants to visit every house a circular road with $n$ houses. For this he starts at a house and then moves clockwise, always skipping exactly $i$ houses before stopping at the next house. For which values of $i$ does he visit every house? I am assuming that $i<n$

I think we need to use residual systems. But I cannot figure out how. For example, say we have $42$ houses. One residual system of $42$ is $\{1,2,3....42\}$. But this does not give use anz new information. of course, I know that $\phi(42) = 12$. But this does not help me either. Are these values actually useful? Or do I need to use another method?

Note: I would prefer methods that do not involve brute force, or rather any kind of program based computation (But if you have a method, please share it anyways.)

For which values of $i$ does he visit every house? $i=1$, I guess — Lucky Chouhan, Sep 29 '23 at 07:38
@LuckyChouhan No, if he he skips 1 house everytime, he will end up skipping half of the houses if $n$ is even — Naitik Mundra, Sep 29 '23 at 07:43
By the dupe: $\bmod n!:\ x\to a+(i+1)x$ is surjective (onto) iff $i+1$ is coprime to $n\ \ $ — Bill Dubuque, Sep 29 '23 at 12:23
Sorry @BillDubuque, did not know about the dupe. I still do not know how I would have found that by myself, but thanks anyways. — Naitik Mundra, Sep 29 '23 at 14:09

Dominique · Accepted Answer · 2023-09-29T07:51:45.447

3

If both $i$ and $n$ have no prime factors in common, all houses will be visited.

Some explanatory examples:
If $i=6$ and $n=12$, you can easily see that only 2 houses will be visited.
If $i=8$ and $n=12$, you'll end up with 3 houses being visited.
If $i=5$ and $n=12$, you'll end up with all houses being visited.

(By the way, there is no reason for $i<n$)

edited Sep 29 '23 at 07:51

answered Sep 29 '23 at 07:50

Dominique

2,130

1

So the condition is that they must be co-prime? Can you prove it too? – Naitik Mundra Sep 29 '23 at 07:51
@NaitikMundra: yes, indeed. – Dominique Sep 29 '23 at 07:52
Simply said: you are looking for a value $a$, for which you can say that for all values of $x$: $(x \cdot i) \mod n \neq a$. – Dominique Sep 29 '23 at 08:22
Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. It should be $i+1$ is coprime to $n$, not $i\ \ $ – Bill Dubuque Sep 29 '23 at 12:24

For $n$ houses arranged in a circle, a man goes clockwise skipping $i$ houses every time. For what value of $n$ does he visit each house?

1 Answers1