Modulo multiplicative inverse of floating numbers

Question

I have a floating value $k$ and an integer $P$

I want to calculate $(\dfrac{k}{\sqrt5}) \mod P$

How do I calculate it?

PS: I know how to calculate MMI (Modulo Multiplicative Inverse of integer values using Fermat's little Theorem ). I am doing $k\times\sqrt5\times 5^{-1} \mod P$.

Here $5^{-1}$ is MMI of $5$ (modulo $P$). But the overall answer is coming wrong.

Eg : $k=4,\,P=7\,$ Hence, $\,4\times\sqrt5\times 5^{-1} \mod 7 = 1.62472$

Is it correct? I guess NO. So how to calculate it then?

Answer 1

The rule $(a \bmod p)·(b\bmod p)\bmod p=(a·b)\bmod p$ only works for integer $a,b$, since in $$ (a+k·p)·(b+m·p)=a·b+(am+bk+mkp)·p $$ the resulting factor in front of $p$ is usually not an integer if $a,b$ are not integers.

Thus your question has empty meaning, you can't even speak of wrong answers. To avoid that, explain how you would test that the answer obtained is correct.

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You could, of course, specify x as the solution of 0==fmod(sqrt(5)*x-k,p). That is, x=(k+7*m)/sqrt(5) without the requirement of x being integer.

Or, in principle equivalent, 0==fmod(5*x-k*sqrt(5),p).

Or, avoiding the roots, 0==fmod(5*x*x-k*k,p), which actually might have proper integer (remainder class) solutions.