Problem:
How many positive even multiples of less than are perfect squares?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Any even multiple of is a multiple of , so we need to find multiples of that are perfect squares and less than . Any solution that we want will be in the form , where is a positive integer. The smallest possible value is at , and the largest is at (where the expression equals ). Therefore, there are a total of possible numbers.
An even multiple square of can be represented by
where is the multiple or and makes it even. Simplifying we have . We can divide by (floor) and get . We can then see that there are different values for . It can't be larger or else .
The problems on this page are the property of the MAA's American Mathematics Competitions