Equation integer part

Jan 2015
39
1
italy
what are all the steps to solve this equation?

97645643 – 6X * lower part( 97645643 / 6X)=X

thank you
 

Benit13

Math Team
Apr 2014
2,166
739
Glasgow
What do you mean by "lower part"?

Is it this?

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

or this?

\(\displaystyle 97645643 - \frac{6x}{\left(\frac{97645643}{6x}\right)} = x\)
 
Jan 2015
39
1
italy
is the first but
for "lower part" means integer part
 

Hoempa

Math Team
Apr 2010
2,780
361
Or maybe \(\displaystyle 97645643 – 6x \cdot \lfloor \frac{97645643}{6x}\rfloor=x\)?

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

Math Team
Apr 2010
2,780
361
Geogebra shows lots of solutions.
 
Jan 2015
39
1
italy
I would like to understand the proceedings
 

Hoempa

Math Team
Apr 2010
2,780
361
It seems like an interesting function. Solutions are \(\displaystyle z \in \mathbb{z} \implies x = \frac{97645643}{6\cdot z+1}\)

Let \(\displaystyle f(x) = 97645643 – 6x \cdot \lfloor \frac{97645643 }{ 6x} \rfloor =x\) 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.