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

[yetam60389]

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!

 

[PeterDonis]

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.

 

[yetam60389]

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.