248. Cats and Mice. One morning at Professor Rackbrain's table, the lively discussion was about the extermination of rodents when suddenly the professor said:

- If a certain number of cats ate a total of 999,919 mice, and all the cats ate the same number of mice, how many cats were there in total?

Someone suggested that perhaps one cat ate all the mice, but Rackbrain objected, saying he had said "cats." Then someone else ventured to suggest that each cat had eaten one mouse, to which the professor remarked that he had said "mice." He also added, to help those present, that each cat had eaten more mice than there were cats.

What, then, would be the correct answer?
248. It is clear that 999919 cannot be a prime number and that, since we need to find a unique solution, it must factorize into a product of two prime factors. These factors are 991 and 1009. We know that each cat caught more mice than there were cats. Therefore, there were 991 cats, and each of them caught 1009 mice.