document.write( "Question 977107: In the regional power station there are a thousand switches on the main console and these are numbered in order of 1-1000. An engineer is employed to control the switches. At the beginning of the shift, the engineer switches on all of the 1000 switches. Sometime later he switches off all the switches with even numbers. A little while later he changes the state of every 3rd switch (3, 6, 9, etc). In other words he changes the switch to “ON” if it was “OFF” and vice versa. Still later, he changes the state of every 4th switch, then every 5th switch and so on until the end of his shift by which by which time has visited the main console 1000 times in all to attend to the switches.
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Algebra.Com's Answer #598609 by Edwin McCravy(20059)\"\" \"About 
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document.write( "1. if he changes the state of a switch an even number of times, the switch will\r\n" );
document.write( "be off, because it was off when he started. \r\n" );
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document.write( "2. if he changes the state of a switch an odd number of times, the switch will\r\n" );
document.write( "be off, opposite from what it was when he started.\r\n" );
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document.write( "3. He will change the state of a switch once for each factor of the number of\r\n" );
document.write( "the switch.\r\n" );
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document.write( "EXPLANATION:\r\n" );
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document.write( "For instance, switch 27 will be turned on when he turns every switch on.  That's\r\n" );
document.write( "once.  Then it will be turned off when the employee changes every 3rd switch,\r\n" );
document.write( "turned on when he changed every 9th switch, and turned off when he changes every\r\n" );
document.write( "27th switch.  That was 4 times, because 27 has the four factors 1,3,9,27.\r\n" );
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document.write( "4. Perfect squares have an odd number of factors, while all other integers have\r\n" );
document.write( "an even number of factors.  That's because factors come in pairs, one factor\r\n" );
document.write( "pairs with another factor produced by dividing the number by that factor.\r\n" );
document.write( "However perfect squares have a square root, which will not form a pair, because\r\n" );
document.write( "it will \"pair\" with itself, which will only gives 1 extra factor besides the\r\n" );
document.write( "pairs. \r\n" );
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document.write( "EXPLANATION:\r\n" );
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document.write( "For instant 25 has factors 1,5,25.  So switch 25 will be turned on when he turns\r\n" );
document.write( "every switch on, then it will be turned off when the employee changes every 5th\r\n" );
document.write( "switch, and turned on again when he changed every 25th switch. That was 3 times,\r\n" );
document.write( "because 25 has the three factors 1,5,25.\r\n" );
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document.write( "5.  Conclusion: the only switches that will be turned on at the end will be the\r\n" );
document.write( "switches whose numbers are perfect squares.\r\n" );
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document.write( "Switches, 1,4,9,16,25,36,... will be on and all the others off.\r\n" );
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document.write( "Edwin
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