A building has three different staircases, all starting at the base of the building and ending at the top. One staircase has 104 steps, another has 117 steps, and the other has 156 steps. Whenever the steps of the three staircases are at the same height, there is a floor. How many floors does the building have?
Solution

Let's call the three staircases $A, B$, and $C$, which have 104, 117, and 156 steps, respectively. Let $a$ be the number of steps in staircase $A$ between each two floors, $b$ be the number of steps in staircase $B$ between each two floors, and $c$ be the number of steps in staircase $C$ between each two floors. By dividing the total number of steps in a staircase by the number of steps that this staircase has between each two floors, we get the number of floors! Therefore,

$$
\frac{104}{a}=\frac{117}{b}=\frac{156}{c}=d
$$

where $d$ is the number of floors in the building. That is, $d$ is a common divisor of 104, 117, and 156. Moreover, $d$ must be the largest possible, as $a, b$, and $c$ are the smallest possible! Therefore, $d$ is the greatest common divisor (GCD) of 104, 117, and 156. Calculating the GCD of 104, 117, and 156, we get 13 as the answer. Therefore, the number of floors in this building is 13.