Final answer:
To make 378#96 divisible by 8, the '#' must be replaced by an 8 to form the number 378896.
Step-by-step explanation:
The question is asking to find the smallest digit that must be placed in the '#' position to make the number 378#96 divisible by 8. To solve this problem, we need to remember that a number is divisible by 8 if its last three digits form a number that is divisible by 8. So, we will focus on the last three digits of the given number, which are '#96'.
Let's work step by step. First, we'll find the possible three-digit numbers ending in 96 that are divisible by 8. We can list out a few numbers like 296, 396, up to 896 and check their divisibility by 8. These three-digit numbers are found to be divisible by 8 if the last two digits (when considered as a 2-digit number) are divisible by 8 or if 8 times a certain number ends with these two digits, 96 in this case. Upon checking, we find that 896 is divisible by 8. Thus, the smallest number we can assign to '#' to make 378#96 divisible by 8 is 8.
Hence, the answer to the question is the number 8.