A1. The national computing competition can have a maximum of 30 participants. This year, the participants solved 4 problems, with $\frac{1}{3}$ of the participants solving exactly 3 problems, $\frac{1}{4}$ of the participants solving exactly 2 problems, $\frac{1}{6}$ of the participants solving exactly 1 problem, and $\frac{1}{8}$ of the participants not solving any problems. How many participants solved all 4 problems?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
A1. The least common multiple of the numbers $3, 4, 6$ and 8 is 24. The number of participants in the competition must therefore be divisible by 24. Since all multiples of the number 24, except for 24 itself, are greater than 30, there were 24 participants in the competition, of which $\left(1-\frac{1}{3}-\frac{1}{4}-\frac{1}{6}-\frac{1}{8}\right) \cdot 24=$ $\frac{3}{24} \cdot 24=3$ participants solved all 4 problems.