Problem:
The increasing sequence consists of those positive multiples of that are one less than a perfect square. What is the remainder when the term of the sequence is divided by ?
Solution:
A positive integer that is one less than a perfect square is of the form
for . Such a number is a multiple of if and only if is not a multiple of . Thus the and terms of the sequence are and respectively. Therefore, the term of the sequence is
When this number is divided by , the remainder is .
The problems on this page are the property of the MAA's American Mathematics Competitions