At the start of a new grading period, Ms. Zhong likes to present her pre-algebra class with extra challenging research problems. One time she wrote the following expression on the chalk board:
Then she turned to the class and said: “In this expression you are to replace the variables with the digits from 1 to 9. Then find its simplified value. Your objective, however, is to substitute in such a way that the value turns out to be an integer, that is, a whole number.”
Her students didn’t waste a moment, setting right to work with their paper, pencils, and, yes, their calculators. In no time at all, Andrei raised his hand, and announced proudly, “I’ve got an answer! I found 2283.”
“Very good work, Andrei,” said Ms. Zhong as she examined his calculations.
Just then Trisha said excitedly, “And I’ve got one bigger than Brian’s number.”
After looking at her paper, Ms. Zhong smiled and said, “You certainly do. Nice going.”
From across the room, Timothy, who was often quiet and shy in math class, cautiously proclaimed, “Teacher, I think I have a smaller answer than they do.”
Ms. Zhong went to his desk, patted him on his head as she grinned from ear to ear, saying, “You really do have a nice number there. I’m proud of you all. Now everybody continue. It seems there are lots of different possibilities. This certainly is a power-full problem, isn’t it?”
By the end of the week the class had found quite a long list of integral values. How many can you find?