# How to find $x$ such that $\require{mathtools}$ $\lceil\frac{n}{b}\rceil \leq\frac{n+x}{b}$ for any $n,b,x\in\mathbb{N}$

Is there a number $x$ such that $$\lceil\frac{n}{b}\rceil \leq\frac{n+x}{b}$$

For example, we can have:

$$\lceil\frac{10}{3}\rceil \leq \frac{10+2}{3}=\frac{12}{3}$$

We can have:

$$\lceil\frac{20}{7}\rceil \leq \frac{20+1}{3}=\frac{21}{3}$$

We can have:

$$\lceil\frac{7}{3}\rceil \leq \frac{7+2}{3}=\frac{9}{3}$$

What is the way to find this $x$ such that we have $\lceil\frac{n}{b}\rceil \leq\frac{n+x}{b}$

If I gave you an arbitary $n$ and $b$, how would you find $x$?

• $b\lceil\frac{n}{b}\rceil-n$? – Marja Aug 7 '17 at 19:33

## 1 Answer

Think about what the left hand side does: It rounds $n/b$ up to the nearest integer.

Now think about what the right hand side does: It just computes $n/b$ and then increases it by $x/b$.

So the difference (not accounting for the increase) in the outputs is less than $1$. This should suggest a good choice of $x/b$, or alternatively a good choice of $x$, given $b$.

But note that $x$ must depend on $b$; if it's too small relative to $b$, it will not work.