Only 1's

For any number with n 3's, the smallest multiple of that number that only has 1's have 3n ones. So the answer is a number with 30 n's.

For an $n$ -digit number $3333...3$ , the multiple that has all the digits equal to $1$ each is a $3n$ -digit number $1111...1$ . In this case $n=10$ , and so the required answer is $3\times 10=\boxed {30}$ .