SOLUTION: how many numbers are there between 500 and 1000 which are divisible by 2 as well as by 3 and whose square roots are whole numbers?

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Question 1074443: how many numbers are there between 500 and 1000 which are divisible by 2 as well as by 3 and whose square roots are whole numbers?
Answer by math_helper(2461) About Me  (Show Source):
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So we're looking for perfect squares between 500 and 1000, divisible by 2 as well as by 3 (—> which means divisible by 6). Let's scan the "square roots" +sqrt%28n%29+ that correspond to +500%3C=n%3C=1000+. Note that for n to be divisible by 6, +sqrt%28n%29+ must also be divisible by 6, because we are considering squares whose roots are whole numbers (no irrationals):
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22^2 = 484 <<< Too small, outside of range
23^2 = 529 <<< 529 is the first candidate, but it is not divisible by 2 or 3
24^2 = 576 (576 is one number)
(skip ahead to 30, because its the next number divisible by 6)
30^2 = 900 (900 is another number)
(skip ahead to 36)
36^2 = 1296 > 1000, too big, outside of range (we're done)
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Ans: Two numbers between 500 and 1000 are divisible by 2 and 3 and have whole number square roots. Those two numbers are 576 and 900.
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