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I understand that the linearity of a function is determined by the degree of the polynomial but I was unsure whether the modulus operator changes this?

Is $f(x) = N \mod x$ a linear function if $N$ and $x$ are integers?

As in:

$f(x) = 17 \mod x$

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    @PeterT.off , $17\ mod\ 1.23\ =\ 1.01$ . $17-1.01\ =\ 15.99.\ \ \ 15.99/1.23\ =\ 13$2012-04-14

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You have to decide what you mean by linear before you can answer this question. The function $f(x)=mx+b$, which you call "definitively linear", satisfies $f(r-s)-2f(r)+f(r+s)=0$ for all $r,s$. The function $f(x)=17$ reduced modulo $x$ doesn't: $f(2)-2f(3)+f(4)=1-4+1=-2\ne0$ If you want to call it linear, go ahead, but beware that it won't do most of the things that you might expect linear functions to do.

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I would say that it is linear over any subset of the domain of the form $(a,a+17)$, where $a$ is where the $ 17 \mod a = 0$.