<center>https://i.imgur.com/VDYtVVH.png</center>

This question is slightly more challenging, hence the red logo. It requires some algebra and number theory but is certainly doable by a secondary level student considering a national pre-Olympiad mathematics competition.

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### Question

Let f(n) = 5n<sup>13</sup> + 13n<sup>5</sup> + 9kn

Find the least positive integer k such that 65 divides f(n) for every integer n.

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The question is taken from the IrMO 2000 paper, that also gives permission to distribute questions so that interested people may openly discuss solutions. It was also featured on my blog GiftedMathematics.com but received no valid answers.

Let's see if Steemians can do better!

As complicated algebra is horrible to write online, I'll accept a logical sequence of key steps in the solution.

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The first correct answer and further interesting comments will be rewarded with an upvote.

Enjoy!

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More Brainsteems currently live:

[Brainsteem Mathematics Challenges: Fibonacci 2018](https://steemit.com/mathematics/@rycharde/brainsteem-mathematics-challenges-fibonacci-2018)

 
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Please Comment, Resteem and Upvote. Thanks!

@rycharde manages the [AAKOM project](https://steemit.com/@aakom) and the [MAP forum](https://steemit.com/@accelerator).

Also check out the new [MAP Rewarder](https://steemit.com/mapforum/@accelerator/introducing-the-new-map-rewarder-passive-income-for-delegators-doc-1-0) for passive income!</center>