Henry Ernest Dudeney/Puzzles and Curious Problems/73 - The Tube Stairs/Solution

by : $73$

 * The Tube Stairs

Solution

 * $839$ steps.

Proof
Let $n$ be the number of steps in the tube station.

Let $1$ be added to $n$ to make $m$.

We have that:
 * $m$ is divisible by $2$
 * $m$ is divisible by $3$

and so on, until:
 * $m$ is divisible by $7$

Thus $m$ is divisible by the lowest common multiple of $\set {2, 3, 4, 5, 6, 7}$.

Hence we calculate:

and we note in passing that $420 = 21 \times 20$ and so is also divisible by $20$

As we are told there are nearly $1000$ steps, it is clear that $420$ is too small for $m$, so we multiply it by $2$ to get $840$.

We then note that $3 \times 420$ is way over $1000$ so cannot be the value for $m$.

So the only possible value for $m$ is indeed $840$.

Hence:
 * $n = 840 - 1 = 839$