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If :$A=1!2!\cdots 1002!$, and $B=1004 ! 1005!\cdots2006!$, how to prove that:

a) $2AB$ is a perfect square

b) $A+B$ is not a perfect square

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    Hint: The exponent of a prime $p$ in $n!$ is $\lfloor\frac np\rfloor + \lfloor\frac n{p^2}\rfloor+ \lfloor\frac n{p^3}\rfloor+\ldots$.2012-09-16

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