document.write( "Question 550545: Find the number of integers less than 2011 that has only 3 factors. \n" ); document.write( "
Algebra.Com's Answer #358983 by Edwin McCravy(20059)\"\" \"About 
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'Find the number of integers less than 2011 that has only 3 factors.
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document.write( "Theorem:\r\n" );
document.write( "A positive integer has exactly three factors if and only if it is\r\n" );
document.write( "the square of a prime number.\r\n" );
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document.write( "If p is a prime number then pē has exactly three factors 1, p, and pē.\r\n" );
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document.write( "Assume for contradiction that n is not the square of a prime but has \r\n" );
document.write( "exactly 3 factors.  If it is the square of a composite integer then it\r\n" );
document.write( "has more than 3 factors, and we have a contradiction.  So n is not a\r\n" );
document.write( "perfect square.  Let k be the third factor other than 1 and n.   \r\n" );
document.write( "Then \"n%2Fk\" is also a factor, and \"n%2Fk\" is not equal to k\r\n" );
document.write( "since n is not a perfect square.  So n has more than 3 factors, and\r\n" );
document.write( "a contradiction is reached.\r\n" );
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document.write( "So we must find the largest square of a prime that does not exceed 2011.\r\n" );
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document.write( "Its prime square root must not exceed the square root of 2011 which is\r\n" );
document.write( "44.84417465\r\n" );
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document.write( "So the primes less than that are\r\n" );
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document.write( "2,3,5,7,11,13,17,19,23,29,31,37,41,43\r\n" );
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document.write( "There are 14 of them and all their squares are the only integers less\r\n" );
document.write( "than 2011 which have exactly 3 factors.\r\n" );
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document.write( "Answer 14\r\n" );
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document.write( "Edwin
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