How many $(n ; k)$ number pairs are there for which $n>k$, and the difference between the interior angles of the $n$-sided and $k$-sided regular polygons is $1^{\circ}$?
Solution. It is expedient to turn to the exterior angles instead of the interior angles, since these are more easily handled, being the sum of the exterior angles of a convex polygon $360^{\circ}$, and on the other hand, the difference of the sought pairs is clearly $1^{\circ}$. Then we are looking for the number of solutions to the equation

$$
\frac{360}{k}-\frac{360}{n}=1
$$

where $3 \leq k<n$ are integers.

Since the second term of the difference is positive, the minuend is definitely greater than 1. If then $m=360-k$, then $0<m<360$ is a positive integer. The equation (1) then takes the form

$$
\frac{360}{360-m}-\frac{360}{n}=1
$$

from which, after rearrangement, we get the equation

$$
m(n+360)=360^{2}
$$

from which it follows that $m$ is a divisor of $360^{2}$.

Conversely, if $0<m<360$ and $m \mid 360^{2}$, then $m<358$ (it is easily verifiable that the largest possible value of $m$ is 324) and then with the choice $3 \leq k=360-m, n=k \cdot \frac{360}{m}$

$$
\frac{360}{k}-\frac{360}{n}=\frac{360}{k}-\frac{m}{k}=\frac{360-m}{k}=1
$$

From this, of course, $k<n$ also follows. It is also obvious that the different divisors of $360^{2}$ less than 360 provide different $(k ; n)$ pairs, so to solve the problem, we need to find the number of divisors of $360^{2}$ that are less than 360.

Factoring $360^{2}=2^{6} \cdot 3^{4} \cdot 5^{2}$. The number of divisors is thus, by the known method, $(6+1)(4+1)(2+1)=105$. Excluding 360, the remaining 104 divisors can be arranged into 52 pairs in the known way, such that the members of each pair are different and their product is $360^{2}$. Thus, for each pair, exactly one of the members is less than 360, the number of the sought divisors is 52, which is also the answer to the question of the problem.

Remark. It can be verified that in the smallest $(k ; n)$ pair, $k=36$ and $n=40$ (the former interior angle is $350^{\circ}$, the latter is $351^{\circ}$), and in the largest pair, $k=359$ and $n=129240$ (the former interior angle is $\approx 358.997^{\circ}$, the latter is $\approx 359.997^{\circ}$).