You forget MoeBlee, that we don't know if some number $\displaystyle 10^{10^{10}}$ might not be an n such that n = 3j + 8k. That is the point of induction. Therefore, while it isnecessaryfor 13 to fail to be such an n, that is notsufficient. By showing that every other n > 13 can be expressed as n = 3j + 8k, we show sufficiency.