Can you install cydia on factory unlock iphone?

I'm in the version 6.1.3.

I don't think there's an unteathered jailbreak for 6.1.3 but if you find one, you can Install it, then if you want to remove it just back up and restore, and you will have your factory unlocked clean iphone back.

Yes, but the more important thing is the status of a jailbreak. Running 6.1.3 is unlikely to have one at all depending on which device you are talking about.

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