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I came across the following question: Let $f(x)=\cos(nx) \sin(\frac{5x}{n})$ have a period of $3\pi$. Find the integral value of $n$

The traditional way of solving this is to equate $f(x) with f(x+3π)$ and then proceed.

However I find this way to be cumbersome and not so fun to do.

We are aware of the facts like $f(p)+f(q)$ will have a period which will be the L.C.M. of the individual periods,

$af(x)$ will have it's period unaffected by the constant a,

And so on

Is there any way I can find the period of two functions multiplied with each other in these forms?

jimjim
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DatBoi
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    do you mean to ask if the period of two functions $f_1$ and $f_2$ are $p_1$ and $p_2$ can we express period of $f_1f_2$ in terms of $p_1$ and $p_2$ ? – jimjim May 04 '20 at 14:53
  • @Arjang exactly – DatBoi May 04 '20 at 14:55
  • just try with two functions with periods 2 and 3, what do you think the period of their product be? also try with two functions with periods 2 and 4, what do you think the period of their product be? looks like something to do with the periods being relatively prime to each other or not. – jimjim May 04 '20 at 15:03
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    @Arjang I couldn't find any general pattern – DatBoi May 04 '20 at 15:04
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    It's not true that the (minimal positive) period of a sum of two periodic functions (with commensurable periods) is the least common multiple of the periods. It can be much smaller. – Daniel Fischer May 04 '20 at 15:08
  • Draw some cases, make some conjectures and ask your observation about the conjuncture. from your question you are already 80% of the way to the solution. before looking at an answer use some pen and paper and play around with ideas. – jimjim May 04 '20 at 15:11
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    @Arjang lemme try some more cases... – DatBoi May 04 '20 at 15:13
  • @Arjang I couldn't generalize it – DatBoi May 04 '20 at 15:53
  • I tried it myself, ran into same problem, it was not as straight forward as I thought, looks like this might help though : https://math.stackexchange.com/a/164238/3936 – jimjim May 04 '20 at 23:53
  • @Arjang thanks! This reference helped me a lot – DatBoi May 05 '20 at 00:58

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