Balance redox reactions in acidic or basic medium.
Combining two half reactions requires the electrons to cancel exactly, so each half must be scaled until both carry the same electron count — the lowest common multiple of the two counts. Permanganate reduction takes five electrons and iron(II) oxidation gives one, so the LCM of five is reached by multiplying the iron half by five. Oxygen and hydrogen are then balanced with water and protons, with hydroxide added at the end in basic solution.
Electron balancing by lowest common multiple
electrons exchanged = lcm(gained, lost); oxidant multiplier = lcm / gained; reductant multiplier = lcm / lost
Because free electrons do not appear in an overall ionic equation. Every electron released by the reducing agent must be taken up by the oxidising agent, so any mismatch means the equation is not balanced for charge.
From the solvent. In aqueous acid, water supplies oxygen and H+ supplies hydrogen, which is why the balanced equation legitimately contains species that were not in the original reagent list.
Because balancing oxygen and hydrogen with water and H+ is mechanical and hard to get wrong. Neutralising the H+ with an equal amount of OH- on both sides at the end converts the result to the basic form without redoing the work.