Coloring each k-th unit in a circle of n units

In summary, the conversation discusses the process of coloring numbers in a circle and determining the number of colored numbers based on the values of n and k. It is mentioned that if n and k are coprime, then the entire circle will be colored. If k divides n, then the number of colored numbers is equal to n divided by k. However, it is noted that this solution may not work for all values of n and k, and further analysis is needed to determine the number of colored numbers.
  • #1
yetam60389
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TL;DR Summary
N numbers in a circle, coloring each k-th one.
suppose you write, clockwise, n numbers (or "units", doesn't matter) in a circle. you then color, clockwise, each k-th number. you do this until you've colored all n numbers, or until you've reached an already colored number. let x be the number of colored numbers.
i've figured that if gcd(n,k)=1, if they're coprime, the whole circle is colored.
furthermore, if k divides n, then x=n/k.

n=10, k=7, then x=10

n=10, k=5, then x=2

tho, what if they're not?
n=10, k=8, then x=5. from where does this 5 arise?

i swear I've solved this before but i just can't find the answer within my brain now.

thanks for any help!
 
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  • #2
yetam60389 said:
n=10, k=8, then x=5. from where does this 5 arise?
What is 10 divided by gcd(10, 8)?

You just need to generalize your analysis a bit.
 
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  • #3
PeterDonis said:
What is 10 divided by gcd(10, 8)?

You just need to generalize your analysis a bit.
figured it out in the meanwhile. you're right, i was quite slow in figuring that out.
 

FAQ: Coloring each k-th unit in a circle of n units

What is the purpose of coloring each k-th unit in a circle of n units?

The purpose of coloring each k-th unit in a circle of n units is to create a pattern or sequence that can be visually represented. This can be helpful in understanding mathematical concepts or for creating aesthetically pleasing designs.

How do you determine which unit to color in a circle of n units?

The unit to be colored is determined by the value of k. For example, if k=2, every second unit in the circle will be colored. If k=3, every third unit will be colored, and so on.

Can any pattern be created by coloring each k-th unit in a circle of n units?

No, the pattern that can be created depends on the values of k and n. Some values may result in a repeating pattern, while others may create a unique design. It also depends on the rules or restrictions set by the person coloring the units.

How many different patterns can be created by coloring each k-th unit in a circle of n units?

The number of different patterns that can be created is infinite, as there are an infinite number of possible combinations for k and n. However, some combinations may result in the same pattern.

Are there any practical applications for coloring each k-th unit in a circle of n units?

Yes, coloring each k-th unit in a circle of n units can have practical applications in various fields such as mathematics, art, and design. It can be used to teach mathematical concepts, create aesthetically pleasing designs, or represent data in a visual way.

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