Problem 371

Suppose the smallest number that is divisible by $12$, $13$, $14$ and $15$ is $N$.

Each number that is divisible by $12$
has factors $3$ and $4$, since $3\times4=12$ and $3$ and $4$ have no factors in common.

Similarly, each number that is divisible by $14$ has factors $2$ and $7$.

However, $N$ must have a factor of
$4$ and thus it has a factor of $2$.

$N$ is divisible by $15=3\times5$ and so $5$ is also a factor of $N$ ($3$
was previously included).

The prime number $13$ is also a
factor of $N$.

The factors of $N$ are $3$, $4$, $7$, $5$, and $13$, and so $N=3\times4\times7\times5\times13=5460$
(this is the lowest common multiple of $12$, $13$, $14$ and $15$).

The smallest five-digit number that is divisible by $12$, $13$, $14$ and $15$, is the smallest integer multiple of
$N$ that is greater than or equal to
$10\,000$.

The smallest integer multiple of $N$
that is greater than or equal to $10\,000$ is $2N=2\times5460=10\,920$.

The hundreds digit of the smallest five-digit number that is divisible
by $12$, $13$, $14$ and $15$ is $9$.