(corrected response, thanks to tutor @ikleyn for pointing out the obvious error....)
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The sum of the digits of 2007 is 9, so the number is divisible by 9:
223 is prime, so we have the prime factorization of 2007:
Then
A perfect square factor of the number has to be of the form
Where a and b are even integers, with 0 <= a <= 4014 and 0 <= b <= 2007.
That gives us 2008 choices for a and 1004 choices for b; the number of perfect square factors of the number is then 2008*1004 = 2016032.
The logic and the development of the solution by @greenestamps are ALMOST correct, except the last line,
which should be edited.
The correct form of the last two lines and the correct answer are as follows:
Where a and b are even integers, with 0 <= a <= 4014 and 0 <= b <= 2007.
That give us 2008 choices for a and = = 1004 choices for b;
the number of perfect square factors of the number is then 2008*1004 = 2016032.
Again, the correct answer for the number of perfect square factors of is 2016032.
