Problem:
It takes algebra books (all the same thickness) and geometry books (all the same thickness, which is greater than that of an algebra book) to completely fill a certain shelf. Also, of the algebra books and of the geometry books would fill the same shelf. Finally, of the algebra books alone would fill this shelf. Given that are distinct positive integers, it follows that is
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let the length of the shelf be unit, and let and denote the thicknesses of the algebra and geometry books, respectively. Then
Thus and it is sufficient to solve the first two equations above for . Multiply the first by , the second by , and subtract: . Thus
Note. The problem asserted that are positive integers. One possible set of integer values are , which yield and .
Query. In the solution, have we used the given information that That the constants are distinct? If so, how?