Problem:
Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number?
Solution:
Let and be the two- and three-digit numbers, respectively. It is given that the juxtaposition is nine times as large as the intended product, hence
Because must be positive, the smallest possible value for is . Because must be at least , it follows that , hence that . Thus can only be , and must be , so .
The problems on this page are the property of the MAA's American Mathematics Competitions