document.write( "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? \n" ); document.write( "

Algebra.Com's Answer #689111 by math_helper(2461) $\"\"$ $\"About$

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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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\n" ); document.write( "22^2 = 484 <<< Too small, outside of range
\n" ); document.write( "23^2 = 529 <<< 529 is the first candidate, but it is not divisible by 2 or 3
\n" ); document.write( "24^2 = 576 (576 is one number)
\n" ); document.write( "(skip ahead to 30, because its the next number divisible by 6)
\n" ); document.write( "30^2 = 900 (900 is another number)
\n" ); document.write( "(skip ahead to 36)
\n" ); document.write( "36^2 = 1296 > 1000, too big, outside of range (we're done)\r
\n" ); document.write( "\n" ); document.write( "—
\n" ); document.write( "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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