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# Birds and Trees

Puzzle:

There is a forest with trees. On each and every tree in the forest, exactly the same number of birds are sitting on it (ex. "There are 5 trees and each tree has 5 birds on it"). The birds in the forest is a given number between 200 and 300. How many trees are in our forest(such that there is no other combination of trees and birds to acquire the same product)?

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Solution:

There are 289 birds and 17 trees in the forest (with 17 birds per tree).

We are looking for a number whose square is between 200 and 300. The candidates are 15, 16 and 17. It can't be 15, because 15*15 = 45*5 = 5*45, so again NO unique answer on the number of trees. Similarly it can't be 16, because 16*16 = 8*32 = 32*8. But 17*17 = 289 and there isn't any other pair of numbers whose product is 289. So, for 289 birds there can be only 17 trees in the forest (with 17 birds per tree) and this is the only unique answer available.