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

Puzzles and Curious Problems by Henry Ernest Dudeney: $151$

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?

The cabbie's number was $121$.

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.

$\blacksquare$