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Look at the intermolecular forces present: NH3 has the capacity to form hydrogen bonds because of its H-N bond. Hydrogen bonds, which are the strongest intermolecular forces, also have dipole-dipole and London forces (in descending order of strength of the intermolecular forces). CH4, on the other hand, only has London forces (which ALL molecules have) and is the weakest intermolecular force. As such, NH3 has a higher boiling point.
NH3 can have Hydrogen bonding (between H and N), dipole-dipole (due to NH3 being a polar molecule), and London dispersion forces. CH4 is nonpolar so it only has London dispersion forces. NH3 has more and also stronger intermolecular forces, so it will have a higher boiling point.
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