Problem 9.2. Denis has cards with numbers from 1 to 50. How many ways are there to choose two cards such that the difference between the numbers on the cards is 11, and the product is divisible by 5?

The order of the selected cards does not matter: for example, the way to choose cards with numbers 5 and 16, and the way to choose cards with numbers 16 and 5 - this is the same way.
Answer: 15.

Solution. For the product to be divisible by 5, it is necessary and sufficient that one of the factors is divisible by 5. Let $n$ be the chosen number divisible by 5, then its pair should be $n-11$ or $n+11$, and both chosen numbers must be natural numbers. Clearly, for $n=5,10,40,45,50$ there is only one way to choose a pair, while for $n=15,20,25,30,35$ there are exactly two ways to choose a pair. In total, $5 \cdot 1+5 \cdot 2=15$ ways.