Problem 3. At a training, there were 225 children and 105 balls. The children were divided into groups with an equal number of children, and the trainers distributed the balls to each group so that each group received an equal number of balls. How many groups were formed and how many balls were distributed to each? How many solutions does the problem have?
Solution. First, we find the common divisors of the numbers 225 and 105, which are the numbers $1, 3, 5$, and 15. These represent the number of groups formed. Therefore, the problem has 4 solutions: 1 group of 225 students receiving 105 balls, 3 groups of $225: 3=75$ students each, with each group receiving $105: 3=35$ balls, 5 groups of $225: 5=45$ students each, with each group receiving $105: 5=21$ balls, 15 groups of $225: 15=15$ students each, with each group receiving $105: 15=7$ balls.