Yes they can because the catalyst does not take part in the chemical reaction and remains unchanged after the reaction has ended. Biological catalysts are known to be recycled by the bodies in which they work.
So base on the question that states and ask to give the name of one cell type that continues to divide throughout a person's lifetime and also on who does not, with that question, the cell that divide through persons lifetime is a Mitosis or skin cells, an example to this is the skin, and the who doesn't divide is i think the brain cells, it doesn't regenerate
If mitosis occurs correctly, the division of one cell by mitosis produces two cells.