6.4. Find all such four-digit numbers: they are all 83 times the sum of their digits.
6.4. There is only one such four-digit number: $1494=18 \times 83$.

Solution 1: Let the number we are looking for be $x$, and the sum of its digits be $a$. From the problem, we know that $x=83a$. Since the remainder when $x$ is divided by 9 is the same as the remainder when $a$ is divided by 9, $x-a=82a$ must be divisible by 9. This indicates that $a$ is a multiple of 9. Since the sum of the digits of any four-digit number does not exceed 36, $a$ can only be $9, 18, 27$, or 36.
(1) $9 \times 83 < 1000$, which is not a four-digit number;
(2) $18 \times 83 = 1494$, which meets the requirements of the problem;
(3) $27 \times 83 = 2241$, but the sum of its digits is not 27;
(4) $36 \times 83 \neq 9999$, and the sum of its digits cannot be 36.

Solution 2: The four-digit number we are looking for is a multiple of 83 and does not exceed $36 \times 83 = 2988$. There are 24 such numbers. After checking each one, we find that only 1494 meets the conditions of the problem.