10. When less than 100 students perform a group dance, there are two combinations: one is a group of 5 in the middle, with the rest forming groups of 8 around the outer circle; the other is a group of 8 in the middle, with the rest forming groups of 5 around the outer circle. How many students are there at most?
10. 93 students
10.【Solution】This problem actually involves an integer less than 100, which when reduced by 5 is divisible by 8, i.e., leaves a remainder of 5 when divided by 8, and when reduced by 8 is divisible by 5, i.e., leaves a remainder of 3 when divided by 5, and we need to find its maximum value. 13 leaves a remainder of 5 when divided by 8, and a remainder of 3 when divided by 5. The least common multiple of 8 and 5 is 40, and \(13 + 2 \times 40 = 93\), which is an integer that satisfies the conditions, meaning there are at most 93 students.