1. An empty bookcase has 6 shelves, and each shelf can hold no more than 50 books. You want to arrange 60 books in the bookcase so that the same number of books is on each shelf used.

a. List the factors of 60.

b. List the factors of 60 that represent the number of books that could be on each shelf.

c. How many different arrangements of books are possible?

d. What are the possible arrangements of the books?

e. Which arrangement uses all of the shelves?

Answer:

A. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60 are factors of 60. 1 and 60 could be omitted as being trivial.

b. 10 books and 6 shelves if all shelves are used. 12 books and 5 shelves if one shelf is left empty. 15 books and 4 shelves if two shelves are unused. 20 books on 3 shelves. 30 books on 2 shelves.

c. 57 different arrangements.

d. 10 books: 1 arrangement; 12 books: 6 arrangements; 15 books: 15 arrangements; 20 books: 20 arrangements; 30 books: 15 arrangements. Total arrangements=1+6+15+20+15=57 arrangements.

e. 10 books on each of the 6 shelves.