# Problem 4. (2 points)

How many negative numbers are there among the numbers of the form $\operatorname{tg}\left(\left(15^{n}\right)^{\circ}\right)$, where $\mathrm{n}$ is a natural number from 1 to 2019?
Answer: 1009

Solution:

$\operatorname{tg} 15^{\circ}>0$.

$15^{2}=225 ; \operatorname{tg} 225^{\circ}>0$.

Further, $225 \cdot 15=3375$, this number gives a remainder of 135 when divided by $360 . \operatorname{tg} 135^{\circ}<0$.
$135 \cdot 15=2025$, this number gives a remainder of 225 when divided by 360. The sequence of remainders when divided by 360 then repeats, meaning that positive and negative tangents will alternate.

The first number does not enter this cycle and it is positive. Among the remaining numbers, exactly half are negative, that is 1009.