### Instruction: A number is divisible by $8$ if the number formed by its last $3$ digits is divisible by $8.$ For example, the number $47\,389\,248$ is divisible by $8$ because $248$ is divisible by $8.$ However, $47\,389\,284$ is not divisible by $8$ because $284$ is not divisible by $8.$

If $992\,466\,1A6$ is divisible by $8,$ where $A$ represents one digit, what is the sum of the possible values of $A?$.### Response: To solve this problem, I need to focus on the last three digits of the given number, which are $1A6.$
I want to find the values of $A$ that make $1A6$ divisible by $8.$
One way to do this is to list all the multiples of $8$ that are between $100$ and $200,$ and see which ones have the form $1A6.$
The multiples of $8$ that are between $100$ and $200$ are: $104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192.$
Among these, the only ones that have the form $1A6$ are $136$ and $176.$
This means that $A$ can only be $3$ or $7.$