56. If the four-digit number $\overline{2 a 17}$ is a multiple of 19, then $a=$
Answer: 7.
Solution: Since the remainder of $2017 \div 19$ is 3, therefore
only when
the remainder of $a \times 100 \div 19$ is 16, $\overline{2 a 17}$ is a multiple of 19.

Also, since
the remainder of $100 \div 19$ is 5,

when $a$ is $1,2,3,4,5,6,7,8,9$ respectively,
the remainders of $100 a \div 19$ are $5,10,15,1,6,11,16,2,7$ respectively,
only when $a=7$, the remainder of $100 a \div 19$ is 16,
that is
$$
2717=19 \times 143 .
$$