4. In a cave under Krim, a terrible beast sleeps. When it gets hungry, it wakes up and eats as many sheep as the sum of the digits of that year. Then it goes back to sleep for as many years as the sheep it has eaten. We know that it woke up on April 12, 354. Could the beast be back by now?

Solve the problems independently. You have 210 minutes for solving.

The use of notes, literature, or a pocket calculator is not allowed.

47th Mathematical Competition

for high school students in Slovenia

Želimlje, April 12, 2003

## Problems for 3rd year students
II/4. Since the number 354 is divisible by 3, the sum of its digits is also divisible by 3, and therefore the number of sheep devoured is divisible by 3. In every subsequent year when the beast awakens, it is thus divisible by 3. The beast cannot awaken in the year 2003, as 2003 is not divisible by 3; therefore, even today, April 12, 2003, it cannot be at the door.