What seat will I be in x steps later?

Question

I need help with the following question regarding modular arithmetic.

"A seating plan consists of 7 chairs in a circle. If I am currently sat in chair 3, what chair will I be in $3^{453}$ steps later?

Assume that for each step, you move one chair forward in the anticlockwise direction and the chairs are numbered from 1 to 7."

Obviously, the number of steps is too large to be computed, so I assume there must be a trick to do this?

Thanks!

Unless you are capable of moving really really really really quickly, you'll probably die before making $3^{453}$ steps... — 5xum, Oct 24 '17 at 14:50

score 0 · Answer 1 · answered Oct 24 '17 at 14:51

0

Hint:

$$3^3\equiv-1\pmod7$$

$(3^3)^{2n+1}\equiv(-1)^{2n+1}=?$

Here $3(2n+1)=453$

answered Oct 24 '17 at 14:51

lab bhattacharjee

274,582

If I solve the equation, I get -1, which I assume means chair 2, since we were at chair 3? Also, how did you get the 2n+1? – Paul H Oct 24 '17 at 14:58
@Paul, As $453/3$ is odd – lab bhattacharjee Oct 24 '17 at 15:00

score 0 · Answer 2 · answered Oct 24 '17 at 15:05

Let's write all the steps down if you want to use modular arithmetic for it.

First you need to test out what happens if you would go 3 steps further in the process and look at the remainder if you would divide it by 7 then you repeat the process from there on till you remark a sequence of all $3^n \pmod 7$ is repeating.

At last you'll try to go through which remainder corresponds with the 453 as the power of 3 $\pmod 7$ then you have the number of steps you will be seated in the seating plan.

Please upvote this answer if you think it was useful – Anonymous196 Oct 24 '17 at 15:51 — Anonymous196, Oct 24 '17 at 15:51

What seat will I be in x steps later?

2 Answers2