3. Two thousand dominoes are arranged in a row at equal (and small) distances from each other. Then, the dominoes are numbered from 1 to 2000. Several more dominoes are lined up in the same way in front of the domino with number 1, and at the other end, $k$ times more $(k>8)$. At the same time and with equal force, the end dominoes are knocked over towards the center. How many dominoes are arranged if, in the end, the domino with number 1984 is left standing?
Solution. Let $x$ be the number of dominos placed in front of the domino with serial number 1; then $16+k x=1983+x$ from which we get $(k-1) x=1967$. Since $1967=7 \cdot 281$, and the numbers 7 and 281 are prime, we have $k-1=281, x=7$, because by condition $k>8$. According to this, the total number arranged is

$$
(k+1) x+2000=283 \cdot 7+2000=3981
$$

dominos.