Based on this problem.

Define two positive integer sequences \(\{a_n\}\) and \(\{b_n\}\) be defined as \(a_1\ne b_1\), \(a_{n+1}=a_n+k\) and \(b_{n+1}=b_n+k\). These two sequences form an **Order \(k\) Divisibility Chain of length \(n\)** if \(a_i\mid b_i\) for \(i=1\to n\).

Prove that no matter what \(n\) and \(k\) you choose, there always exists an infinite number of sequences \(\{a_n\}\) and \(\{b_n\}\) that form an Order \(k\) Divisibility Chain of length \(n\).

Please only post hints, do not post the solution. If you do, it will give away the solution for the problem I based this on. Thanks.

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## Comments

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TopNewestTake \(a_1=k, b_1=k+k\times n!\). This gives an example of a single such sequence for any n,k. Should not be hard to generalize this to produce infinitely many such sequences,

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Hint: \(a_1 = b_1\)

(Which, by the way, is why I reported the problem.)

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I can't believe I didn't realize that any \(a_1=b_1\) works. Edited the problem and this note to show that \(a_1\ne b_1\).

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Hint: Consider the sequence in modulo \(lcm(a_1,a_2,...,a_n)\). From there rest will be pretty easy.

@Daniel Liu : Am I being too obvious?

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No, you're not being too obvious as far as I can tell.

Now to think about it, I should have posted this problem for the Proofathon Sequences and Series competition. Dang!

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