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I'm trying to follow a proof for the solidarity of the periodicity of a communicating class of a time-homogenous discrete time Markov chain in my course notes for applied probability and part of it requires the following to hold:

Suppose $W$ is a set of integers with the property that

$$m,n \in W \implies m+n \in W.$$

Define $d$ to be the greatest common denominator (gcd) of all elements of $W$.

Then $nd \in W$ for all sufficiently large $n.$

I understand the definition of $W$ and $d$, I just don't understand how it implies $nd \in W$ for all sufficiently large $n.$

I've tried to use Bézout's identity and the fact that the gcd of the set of all the elements of $W$ divided by $d$ is $1$. Also not sure if it actually holds when all elements of $W$ are negative (as $d$ would still be positive?? I might just be really over-thinking this, but please help! Thank you!

Sean Roberson
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Hayley Inglis
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    Normally, a math textbook will give you tools in advance, that you can use to verify such a statement. What tools (i.e. worked examples, previously solved problems, established theorems, analytical methods in the proofs of such theorems) were given to you in advance that you think might be pertinent? I suggest editing your posted question by reporting the tools and then demonstrating a significant attempt to attack the problem by using these tools. – user2661923 Jun 19 '25 at 14:50
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    Sorry, but you'd have to provide literal quotes from whatever source you have. I've listened to Kolmogorov at a conference in 1978 or 1979, and he was still brillant. – wasn't me Jun 19 '25 at 14:52
  • I needed to look this up for my own course, and found this question which includes a proof in the question body. Welcome to the site – FShrike Jun 19 '25 at 18:52
  • Unfortunately you have used $n$ in two different places with different meanings, so let's adjust the claim to "Then $kd\in W$ for all sufficiently large $k$." The largest $k$ for which it might not be true is $\dfrac{mn}{d^2}-\dfrac{m+n}{d}$ (if $W$ has elements $m$ and $n$ with hcf or gcd $d$), but it is true for all larger $k$. It is a variation of https://math.stackexchange.com/questions/66963 which has $d=1$. – Henry Jun 20 '25 at 21:49

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