[ Divisibility of numbers. General properties ]

The year of the current mathematics festival is divisible by its number: $2006: 17=118$.

a) Name the first number of the math festival for which this was also true.

b) Name the last number of the math festival for which this will also be true.

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a) The first math festival was in 1990. It is clear that the year of its holding is divisible by its number, because the number is one.

b) Let $N$ be the number of the math festival. Then the year of its holding is (2006 - 17) $+N=1989+N$. Suppose the year of the festival is divisible by its number, that is, $1989+N$ is divisible by $N$. Therefore, 1989 is divisible by $N$. Since we are looking for the largest possible $N$, we should take $N=1989$.

## Answer

a) 1 ; b) 1989.