Problem number 25. Daniel wants to complete the diagram so that each box in the middle and top rows will contain the product of the values in the two boxes below it and each box contains a positive integer. He wants the value in the top box to be 720. How many different values can the integer n take? So first let's make the diagram larger and let's also write a and b in the bottom row. That means that the second row will have a n and n b and the top row will have a n squared b because a n times n b will be a n squared b. So what does this tell us about n? Well it tells us that n squared must be a perfect square divisor of 720 because we know n squared times something equals 720 and this is the only condition on n. Any n such that n squared is a perfect square divisor of 720 will be a valid end to the problem. So that means we just need to count how many n are there such that n squared is a perfect square divisor of 720 or in other words how many perfect square divisors of 720 are there. We can notice that the largest perfect square divisor of 720 is 144 or 12 squared and we can do this just by going down from the divisors of 720. 144 times 5 equals 720. This means we all need to check the perfect squares up to 12 squared and see how many of them are perfect square divisors of 720. For example all of the perfect squares from 1 to 4 squared are divisors of 720. From 5 to 8 squared only 6 squared equals 36 is a divisor of 720 and as shown before 12 squared is divisor 720 but none of 9 to 11 squared are. So this means that the only values of n are 1, 2, 3, 4, 6, and 12. That means that n can take six possible values so the answer is d, 6.

perfect square divisor

value of n

problem solving

720

diagram