# Mathematical Moving Chairs

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 Number of people -> Multiply by ... 6 7 8 9 10 ... 2 . ? . . ? . . ? . . ? . . ? . 3 . ? . . ? . . ? . . ? . . ? . 4 . ? . . ? . . ? . . ? . . ? . 5 . ? . . ? . . ? . . ? . . ? . 6 . ? . . ? . . ? . . ? . . ? . ...
Suppose we have 8 people in chairs numbered 0 to 7. Each person multiplies their seat number by 5, divides by 8 (being the number of people) and keeps the remainder. That's their new seat.

Do they all go to different seats?

• What about with 9 people?
• What about 10 people?
• What about 11 people?
• What if we multiply by 4?
• ... or by 6?
• ... or by 7?
When does it work?

And why?

This is intimately related to Modulo Arithmetic.

One of the enrichment tasks on this web site.
CategoryMaths
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