Suppose we have three random variables $X_1, X_2$ and $X_3$. For pairwise independence it is sufficient to show that $$E[X_{1}X_{2}] =E[X_1]E[X_2]$$, $$E[X_{1}X_{3}] = E[X_1]E[X_3]$$ and $$E[X_{2}X_{3}] = E[X_{2}]E[X_{3}]$$
For mutual independence would it be enough to show that $$E[X_{1}X_{2}X_{3}] = E[X_{1}]E[X_{2}]E[X_{3}]$$?