Carlos is arranging books on shelves. he has 24 novels and 16 autobiographies. each shelf will have the same numbers of novels and autobiographies. if carlos must place all the books on shelves, what are the possible numbers of shelves carlos will use?

You need to calculate the common divisors of both numbers 24 and 16. To do that, factor each number: 24 = 2*2*2*3 and 16 = 2*2*2*2. That means that the common divisors are 2, 2^2 = 4 and 2^3 = 8. Start by using 2 shelves: each of them will have 24/2 = 12 novels and 16/2 = 8 autobiographies. Now use 4 shelves: each shelf will have 24/4 = 6 novels and 16/4 = 4 autobiographies per shelf. Use 8 shelves: 24/8 = 3 novels per shelf and 16/8 = 2 autobiographies per shelf. Of course, he can use just 1 shelf (1 is a divisor of any number) with all the books. So, the answer is tha the possible numbers of shelves Carlos will use is: 1, 2, 4, and 8.