Final answer:
Mariana can distribute her 20 math books and 10 science books evenly among 10 bookshelves by placing 2 math books and 1 science book on each shelf.
Step-by-step explanation:
To find the greatest number of bookshelves Mariana can use to distribute her 20 math books and 10 science books evenly, we must find the greatest common divisor (GCD) of 20 and 10, since the books need to be distributed in such a way that each shelf has the same combination of math and science books without any leftover.
The divisors of 20 are 1, 2, 4, 5, 10, and 20. The divisors of 10 are 1, 2, 5, and 10. The highest common divisor among these numbers is 10. Therefore, Mariana can evenly distribute her books among 10 bookshelves, with each shelf containing 2 math books and 1 science book, ensuring no books are left over and maintaining the combination on each shelf.