Task 10.5. Vika has 60 cards with numbers from 1 to 60. She wants to divide all the cards into pairs so that the modulus of the difference of the numbers in all pairs is the same. How many ways are there to do this?
# Answer: 8.

Solution. Let $d$ be the absolute difference between the numbers. It is clear that the number $d$ is a natural number.

It is clear that the number 1 must be paired with $d+1$, the number 2 must be paired with $d+2, \ldots$, the number $d$ must be paired with $2d$. Therefore, the first $2d$ natural numbers are paired with each other.

Considering the number $2d+1$, we get that it must be paired with $3d+1$, the number $2d+2$ must be paired with $3d+2, \ldots$, the number $3d$ must be paired with $4d$. Therefore, the next $2d$ natural numbers are also paired with each other.

Continuing in this manner, we get that the set of all numbers $\{1,2, \ldots, 60\}$ should be divided into non-overlapping groups of $2d$ numbers (which in turn are divided into $d$ pairs). Therefore, 60 is divisible by $2d$, which is equivalent to 30 being divisible by $d$. The number 30 has exactly 8 natural divisors: $1,2,3,5,6,10,15,30$. It is clear that all of them are suitable, as each number $d$ among them corresponds to $\frac{30}{d}$ groups of the form $\{2kd+1,2kd+2, \ldots, 4kd\}$, which are obviously divided into pairs with the absolute difference $d$.