Problem 1. In a room, there are chairs with 3 legs and chairs with 4 legs. One person is sitting on each chair, so there are 69 feet on the floor. Determine the number of chairs with 3 legs and chairs with 4 legs.
Solution. Let $x$ be the number of chairs with 3 legs, and $y$ the number of chairs with 4 legs. From the condition of the problem, we have $3x + 4y + 2(x + y) = 69$, which simplifies to $5x + 6y = 69$. We seek the solution in the set of natural numbers. Since $6y$ is an even number, the number $x$ must be odd. Multiplying the last equation by 2, we get $10x + 12y = 138$, from which $12y = 138 - 10x$. Therefore, $12y$ must end in 8 and $x$ must be less than 12. From this, we conclude that the problem has two solutions: $x=3, y=9$ or $x=9$, $y=4$.