# Equation integer part

#### gerva

what are all the steps to solve this equation?

97645643 â€“ 6X * lower part( 97645643 / 6X)=X

thank you

#### Benit13

Math Team
What do you mean by "lower part"?

Is it this?

$$\displaystyle 97645643 - 6x\left(\frac{97645643}{6x}\right) = x$$

or this?

$$\displaystyle 97645643 - \frac{6x}{\left(\frac{97645643}{6x}\right)} = x$$

#### gerva

is the first but
for "lower part" means integer part

#### Hoempa

Math Team
Or maybe $$\displaystyle 97645643 â€“ 6x \cdot \lfloor \frac{97645643}{6x}\rfloor=x$$?

(Ah, I see now you mention the integer part thing.)

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#### Hoempa

Math Team
Geogebra shows lots of solutions.

#### gerva

I would like to understand the proceedings

Math Team
Not sure (yet).

#### Hoempa

Math Team
It seems like an interesting function. Solutions are $$\displaystyle z \in \mathbb{z} \implies x = \frac{97645643}{6\cdot z+1}$$

Let $$\displaystyle f(x) = 97645643 â€“ 6x \cdot \lfloor \frac{97645643 }{ 6x} \rfloor =x$$ and let lpf(n) be the smallest primefactor of a positive integer.
So lpf(4) = 2.
For composite numbers c not or the form $$\displaystyle 2^m \cdot 3^n$$, it seems that f(97645643/c) = 97645643/(c/lpf(c)). One might find this helpful to find primefactors. Interesting function you gave.