Solution:

Clearly, the last digit must be $5$ and we have to determine the remaining $6$ digits. For divisibility by $7$, it is sufficient to consider the number obtained by replacing $7$ by $0$; for example $5775755$ is divisible by $7$ if and only if $5005055$ is divisible by $7$. Each such number is obtained by adding some of the numbers from the set $\{50, 500, 5000, 50000, 500000, 5000000\}$ along with $5$. We look at the remainders of these when divided by $7$; they are $\{1, 3, 2, 6, 4, 5\}$. Thus it is sufficient to check for those combinations of remainders which add up to a number of the form $2 + 7k$, since the last digit is already $5$. These are $\{2\}, \{3, 6\}, \{4, 5\}, \{2, 3, 4\}, \{1, 3, 5\}, \{1, 2, 6\}, \{2, 3, 5, 6\}, \{1, 4, 5, 6\}$ and $\{1, 2, 3, 4, 6\}$. These correspond to the numbers $7775775, 7757575, 5577775, 7575575, 5777555, 7755755, 5755575, 5557755, 755555$.