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I did understand however that a lot of network admins don't understand it 100% either, and because of this use some sort of online subnet calculator to find the correct subnet and mask they need for their project. (for the reference: I don't think you're one of those!)

I understand it, but I still use a calculator because the math is sort of irritating. It's actually fairly easy to memorize the majority of it, but I don't use it often enough for it to stick in my memory.
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Bitt Faulk