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I don't understand the conversion from eV/c to kg*m/s

Hi, so I understand that when you divide energy by the speed of light, c, you get momentum, and that's how the units eV/c come about. I get that 1 eV/c = 5.36 x 10^-28 kg*m/s by multiplying by 1.6 x 10^-19 J/eV and dividing by 3 x 10^8 m/s. I'm just struggling to understand how the units work, as the multiplication by 1.6 x 10^-19 J/eV would convert the units into J/c, but then wouldn't multiplication by c turn the units back into units of energy? Except they are still units of momentum- I just can't see how the conversion makes the units equivalent to kg*m/s.
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Krell1
E = MC^2
E = PC (P is momentum)

MC^2 = PC

P= MC (units kg*ms^(-1))

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