Suppose a number $n$ exists with the property that $3n$ and $7n$ have only even digits and let $N$ be the smallest such. If $N$ is divisible by $10$, $\frac{N}{10}$ is a smaller such number. Hence $N$ ends in one of $2, 4, 6, 8$. But $3N + 7N = 10N$ and $10N$ has the tens digit odd as a result of the carry from the unit digits. But this implies $N$ ends in an odd digit which is impossible.