Your smartphone or tablet might look like it's off, but even when the screen is dark it's doing stuff. And that stuff drains the battery. And router configurations can have a huge impact on that power drain.
Just because your smartphone or tablet (or smartwatch for that matter) looks like it's doing nothing, that doesn't mean that there's nothing going on. If it's connected to a network, there are communications going on that constantly wake it up from sleep mode, and eat at your battery life.
These communications come in the form of Router Advertisement (RA) messages, and according to a Internet Engineering Task Force (IETF) memo RCF 7772...
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