Floor function

[trail dock]

essaie d'écrire $\lfloor k\frac{p}{q} \rfloor = k\frac{p}{q} - {k\frac{p}{q}}$

[quaint gulch]

bloubbloub

[trail dock]

où {} désigne la partie fractionnaire

[elder token]

Reminds me of the sum of an arithmetic series. Maybe you can show a relationship with (q-k)(p/q) and kp/q?

I don't speak French, but I assume it says "let"....."be two coprime numbers. Show that".....

[trail dock]

yes

[boreal terrace]

it's just a number of integer points under the line y=px/q over [0,q-1]. So it is half of the number of integer points inside a rectangle p-1 by q-1. There is a slight nuance at the corner, but it has no integer points in it. So, you get your formula

[next bramble]

Thanks 🙏

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