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Batteries do slowly discharge even when a device is turned off. This is an entirely normal occurrence if the Kindle's been switched off for 4 months.
... Having said which, I'm a little puzzled by what you mean when you say that you "shut it down". I don't know of any method for shutting down a Kindle. You can put it to sleep, or you can reboot it, but how do you shut it down?
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