## Subject III. (20 points)

The natural numbers $1, 2, 3, \ldots, 2013$ are written on cards, with the written side facing down. We say that a card is "lucky" if the number written on it is divisible by 20 or 13. What is the smallest number of cards that must be turned over, without looking, to be sure that at least one of them is "lucky"?

Prof. Sorin Borodi, Alexandru Papiu Ilarian Theoretical High School, Dej
## Subject III.

The multiples of 20 less than 2013 are $20 \cdot 1, 20 \cdot 2, \ldots, 20 \cdot 100$, which are 100 numbers.

The multiples of 13 less than 2013 are $13 \cdot 1, 13 \cdot 2, \ldots, 13 \cdot 154$, which are 154 numbers.

Since 20 and 13 are coprime, $\Rightarrow [20; 13] = 20 \cdot 13 = 260$.

(10 points)

The multiples of 260 less than 2013 are $260 \cdot 1, 260 \cdot 2, \ldots, 260 \cdot 7$, which are 7 numbers.

Consequently, the number of "lucky" cards is $100 + 154 - 7 = 247$.

The number of cards that are not "lucky" will be $2013 - 247 = 1766$.

In the worst-case scenario, it is possible that the first 1766 cards turned over are precisely those that are not "lucky"; even in this situation, the next card turned over will be "lucky".

In conclusion, the answer is 1767.

(10 points)