Henry Ernest Dudeney/Puzzles and Curious Problems/165 - The Postage-Stamps Puzzle/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $165$

A youth who collects postage stamps was asked how many he had in his collection, and he replied:

"The number, if divided by $2$, will give a remainder $1$;

divided by $3$, a remainder $2$;

divided by $4$, a remainder $3$;

divided by $5$, a remainder $4$;

divided by $6$, a remainder $5$;

divided by $7$, a remainder $6$;

divided by $8$, a remainder $7$;

divided by $9$, a remainder $8$;

divided by $10$, a remainder $9$.

But there are fewer than $3000$."

Can you tell how many stamps there were in the album?

$2519$.

Let $n$ be the number of stamps in the collection.

We have that $n + 1$ is divisible by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ and $10$.

The smallest such $n + 1$ is $\lcm \set {2, 3, 4, 5, 6, 7, 8, 9, 10}$ which is $8 \times 9 \times 5 \times 7 = 2520$.

No other number smaller than $3000$ fits the conditions.

Hence $n = 2519$.

$\blacksquare$