The facebook computers I log in from are 'not again', 'grrr', 'told you last time' etc - the feature is buggy as hell. I am a laptop user, but even my home ISP was identifying as 'different' depending on which setting you happened to hit, and I'm tired of that game.
As for involving a mobile phone - have you any idea how many people don't own a mobile phone, run out of charge, aren't in reach of a mobile network, are travelling abroad where their phones don't work, forget their phone and want to check something on their work computer, etc etc at any given time? This would cause a major headache for a lot of people, and giving out your phone number to random networks when other networks use your phone number to verify your identity sounds like a very, very bad idea.
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As for involving a mobile phone - have you any idea how many people don't own a mobile phone, run out of charge, aren't in reach of a mobile network, are travelling abroad where their phones don't work, forget their phone and want to check something on their work computer, etc etc at any given time? This would cause a major headache for a lot of people, and giving out your phone number to random networks when other networks use your phone number to verify your identity sounds like a very, very bad idea.