M7MBA wrote: ↑

Thu Dec 17, 2020 11:27 am

\(X\) number of books have to be distributed equally among a number of library shelves. If \(X\) is a multiple of both 9 and 6, which of the following statements must be true?

I. The books can be distributed equally among 3 library shelves.
II. The books can be distributed equally among 27 library shelves.
III. The books can be distributed equally among 54 library shelves.

A. I only
B. II only
C. III only
D. I, II, and III
E None of the above

Answer: A

Solution:

Since X is a multiple of both 9 and 6, we can let X be the least common multiple of 9 and 6, which is 18. If X = 18, we see that the 18 books can be distributed equally among 3 shelves. However, they can’t be distributed equally among 27 or 54 shelves since some shelves will have at least one book while others will have no books. So II and III must be false.

Now, if X is not 18, then it must be a multiple of 18 since X is a multiple of both 9 and 6 (recall that if k is a multiple of m and n, then k is a multiple of the LCM of m and n). Since 18 is already a multiple of 3, any multiple of 18 will be also a multiple of 3; therefore, the number of books can be always distributed equally among 3 shelves. So I must be true.

Answer: A