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Answer: 2008. d(n) is number of divisors of n. I give number of divisors and list them also. The divisors of n = 2008: 1, 2, 4, 8, 251, 502, 1004, 2008 d(2008) = 8 The divisors of n = 2009: 1, 7, 41, 49, 287, 2009 d(2009) = 6

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How many positive integers less than 2008 have an even number of divisors?

There are 1,963 such integers. Every factor of a number has a pair. The only time there will be an odd number of factors is if one factor is repeated, ie the number is a perfect square. So the question is really asking: how many positive integers less than 2008 (in the range 1 to 2007) are not perfect squares. √2007 = 44 and a bit (it lies between 44 and 45) So there are 44 integers less than (or equal to) 2007 which are perfect squares → 2007 - 44 = 1963 integers are not perfect squares in the range 1-2007 and have an even number of factors (divisors).


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