Question

How many non-negative integral values of $x$ satisfy the equation $\left[\frac{x}{5}\right]=\left[\frac{x}{7}\right]$?
(Here $\left[x\right]$ denotes the greatest integer less than or equal to $x$. For example $\left[3.4\right]=3$ and $[-2.3=-3).$

Answer 1

It's clear that as $n$ increases, $\frac{n}{5}$ will eventually be much larger than $\frac{n}{7}$. As long as the difference is at least $1$, the integer parts can't be equal; a (very) little algebra gives us

$\frac{n}{5}\ge \frac{n}{7}+1\to n\ge \frac{7\times 5}{7-5}=\frac{35}{2}$

There is a corresponding result for $n$ negative, but we don’t have to worry about it.

Of course, it doesn't follow that the integer parts of the quotients are the same for all smaller numbers.

$[5\times 1,7\times 1)$

$[5\times 2,7\times 2)$

$[5\times 3,7\times 3)$

are all obviously no good; dividing by $7$ clearly gives a smaller number than dividing by $5$. In the middle of this last range we trip over the absolute limit we found before.

Notice that there are five good values of $n$ in $[0,5)$ (not a coincidence), and three good values in the next continuous group of good values, in $[7,10)$ (decreasing from $5$ to $3$, that is, by $7-5$, again no coincidence). Then there's a lonely individual $n$ in $[14,15)$, and that's it.

So the size of the solution set is $5+3+1=9$.