What is the maximum value that must be assigned to A so that the 8-digit number 816A0408 is divisible by 4?

The correct option is 9.

To determine the maximum value that can be assigned to A so that the 8-digit number 816A0408 is divisible by 4, let us analyze the rule for divisibility by 4.

Divisibility Rule for 4:
A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

In the given 8-digit number, 816A0408, the last two digits are 08.

Let us check if the number formed by the last two digits (08) is divisible by 4:

$\frac{08}{4}=2$

Since 8 is divisible by 4, the number 816A0408 is divisible by 4 regardless of the value chosen for the digit A.

Therefore, A can be any single-digit whole number from 0 to 9, i.e., A ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

Since the question asks for the maximum value that can be assigned to A, we choose the largest single digit among the possibilities:

$Maximum\; value\; of\; A=9$

Thus, the maximum value that must be assigned to A is 9.