Southern Colorado College has between 900 and 1,000 dormitory rooms. Each dormitory has the same number of floors and the same number of rooms per floor. Every room number has three digits, the first of which shows the floor; for example "205 and 211 are the fifth and eleventh rooms on the second floor of every dorm. The total number of rooms in all the dorms is an odd number that is not divisible by 3 or 5. What's the highest room number?

Solution to the Problem: The highest room number is 719. The total number of rooms is the product of the number of dorms, the number of floors in each dorm, and the number of rooms on each floor. Since the total is an odd number, each of these factors must be odd. The number of floors is an odd one-digit number. It can't be 1 because #205 is on the second floor, and it can't be 3, 5, or 9 because any of these would make the total divisible by 3 or 5. So there are 7 floors. The total is between 900 and 1000. The number of rooms on each floor must be an odd two-digit number. The smallest odd two-digit number is 11. The number of dorms can't have three or more digits, because 7 (floors) x 11 (rooms per floor) x 100 (smallest three-digit number) = 7700 (too big). The number of dorms can't be an odd two-digit number because: (1) if the three factors are 7 x 11 x 11, the product is 847, less than 900; (2) if the factors are 7 x 11 x 13, the product is more than 1000; and (3) any other combination will have a still larger product. So the number of dorms must be an odd one- digit number. For the same reasons as with the number of floors, it must be 7. There are 7 dorms and 7 floors. Seven times 7 is 49. The only number of rooms per floor that can be multiplied by 49 to produce an odd total between 900 and 1000 is 19. So the largest room number is 719.

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