# Closed form solution for $\left \lfloor{\frac{a}{x}}\right \rfloor$ = $\left \lfloor{\frac{b}{x}}\right \rfloor$?

I need to find the smallest value of $x$ such that:

$\left \lfloor{\frac{a}{x}}\right \rfloor$ = $\left \lfloor{\frac{b}{x}}\right \rfloor$

EDIT: where $0 < x < a < b$, and $x \in \mathbb{N}$

Is there a closed-form solution for this problem? Any help or suggestions are greatly appreciated.

Thanks.

• hint: if $0<a<x<b$ the equality is invalid; but if $0<a<b<x$ the result is correct. – Hilario Fernandes Oct 15 '14 at 23:37
• @HilarioFernandes But the inequality may be valid when $0<x<a<b$, for example, $x=3$, $a=6$, $b=7$. – user21467 Oct 16 '14 at 0:12
• There is often no such smallest $x$; for example, $\lfloor\frac2x\rfloor < \lfloor\frac3x\rfloor$ if $0<x\le\frac32$ and $\lfloor\frac2x\rfloor = \lfloor\frac3x\rfloor$ if $\frac32<x\le2$. Do you want the infimum? Or do you want $\lceil\cdot\rceil$? – user21467 Oct 16 '14 at 15:10
• @StevenTaschuk My problem was ill defined. In fact, $x \in \mathbb{N}$. – mandy Oct 16 '14 at 20:37
• @WillJagy I wish it was an assignment! I am implementing a big data algorithm where I could really benefit from the speed-up of a closed form solution. So far, I iterate between $b-a+1<x<a-1$ but that's not viable for very large values of $a$ and $b$. – mandy Oct 17 '14 at 21:31

x = 1

a = 1/2

b = 3/4

These three values satisfy the equality.

$$\lfloor \frac ax \rfloor = \lfloor \frac bx \rfloor$$

Substituting in the values...

$$\lfloor \frac 12 \rfloor = \lfloor \frac 34 \rfloor$$

$$0 = 0$$

The smallest value for x is 1.

a and b can be any numbers in the range [n, n+1) where n is an integer so long as b > a. Their solution is infinite.