What is the mass number of an ion with 105 electrons, 157 neutrons, and a 1 charge?

Alright, first you have to understand that the mass number is simply the number of neutrons + the number of protons.

However, they've only given you the electrons and the neutrons. Assuming that you meant "1 charge" and "+1 charge" that means you've got 106 protons and 157 neutrons.

Mass number = neutrons + protons
                       = 106+157
                       = 263