Henry Ernest Dudeney/Puzzles and Curious Problems/151 - The Arithmetical Cabby/Solution

by : $151$

 * The Arithmetical Cabby
 * The driver of the taxi-cab was wanting in civility, so Mr. Wilkins asked him for his number.
 * "You want my number, do you?" said the driver.
 * "Well, work it out for yourself.
 * If you divide by number by $2$, $3$, $4$, $5$, or $6$ you will find there is always $1$ over;
 * but if you divide it by $11$ there ain't no remainder.
 * What's more, there's no other driver with a lower number who can say the same."


 * What was the fellow's number?

Solution
The cabbie's number was $121$.

Proof
Let $n$ be the driver's number.

We know that $n - 1$ is divisible by $2$, $3$, $4$, $5$ and $6$.

Hence we know that:
 * $n - 1 = k \times \lcm \set {2, 3, 4, 5, 6} = 60 k$

where $k$ is an integer.

We see immediately that:


 * $k = 0 \implies n = 1$

which is not divisible by $11$


 * $k = 1 \implies n = 61$

which is not divisible by $11$


 * $k = 2 \implies n = 121$

which is $11 \times 11$ and so is the number we want.