Problem:
What is the largest -digit prime factor of the integer
Solution:
Let be a prime less than . Then
Thus, if in addition , we have
Now, such a prime divides iff . Once we show that there is at least one prime meeting these conditions, our answer will be the largest such prime, because any with will be smaller. Since , the requirement that be as large as possible leads to our choosing as small as possible. Thus we take and , for which the largest is . Since , the second display above is correct for . Thus divides and is the largest -digit prime to do so.
The problems on this page are the property of the MAA's American Mathematics Competitions