10 These numbers are the square numbers: 12, 22, 32, and 42 These numbers are the square numbers: 12, 22, 32, and 42. So, there is an open locker door at every square number. How many square numbers are there between 1 and 1000? Through a little trial and error, you'll find that 312 is the last square number less than So, there are 31 open doors (the last one occurring on the door numbered 312 or 961).
11 Looking at Factors A special property of square numbers is that they always have an odd number of factors. A factor is a number that divides another number evenly (with no remainder). For example, 8 has an even number of factors, namely, 1, 2, 4, and 8. But, 9 has an odd number of factors, namely, 1, 3, and 9. In fact, all numbers except the square numbers have an even number of factors.
12 Student Leaves locker 40 1 Open 2 Closed 4 5 8 10 20 40 You can use this fact to solve the locker problem. Take any locker number, 40, for example. Its state (open or closed) is changed for every student whose number in line is a factor of the locker number. So, write out all the factors of 40, like this:StudentLeaves locker 401Open2Closed458102040
13 Like all other lockers numbered with non-square numbers, it ends up closed after all the students have gone through the line because it has an even number of factors.
14 Here's the factor pattern for a square number, 16. StudentLeaves locker 161Open2Closed4816Locker 16 remains open because it has an odd number of factors.
15 You can now conclude that all the doors with square numbers on them will remain open because all square numbers have an odd number of factors. You can now conclude that all the doors with non-square numbers on them will remain closed because only square numbers have an odd number of factors.
16 There are 31 open doors. The numbers of the open doors are listed below. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961
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