Problem:
Given a positive integer , let be the product of the non-zero digits of . (If has only one digit, then is equal to that digit.) Let
What is the largest prime factor of
Solution:
Consider each positive integer less than to be a three-digit number by prefixing 's to numbers with fewer than three digits. The sum of the products of the digits of all such positive numbers is
However, is the product of the non-zero digits of . The sum of these products can be found by replacing by in the above expression, since ignoring 's is equivalent to thinking of them as 's in the products. (Note that the final in becomes a and compensates for the contribution of after it is changed to .) Hence
and the largest prime factor is .
The problems on this page are the property of the MAA's American Mathematics Competitions