To solve the problem, let’s analyze it step by step:

Problem Details:

We are given a 12-digit number $566221111A42A$, where $A$ is a digit that needs to be determined.
When this number is divided by 4, the remainder is 3.

Rule for Divisibility by 4:

A number is divisible by 4 if the number formed by its last two digits is divisible by 4. For numbers not divisible by 4, the remainder is determined by the last two digits.

Thus, the last two digits here are $42A$, where $A$ is the unknown digit. We need to find $A$ such that: $\text{(Value of 42A)} \mod 4 = 3.$

Step-by-Step Solution:

1. Express the last two digits as $42A$: The number $42A$ can be written as $420 + A$, where $A$ is a single digit (0–9).

2. Check divisibility for $420 + A$: Since $420$ is divisible by 4, the divisibility (or remainder) depends only on $A$. Specifically, we need: $A \mod 4 = 3.$

3. Find possible values for $A$:

   • When $A = 0$, $0 \mod 4 = 0$.
   • When $A = 1$, $1 \mod 4 = 1$.
   • When $A = 2$, $2 \mod 4 = 2$.
   • When $A = 3$, $3 \mod 4 = 3$.
   • When $A = 4$, $4 \mod 4 = 0$.
   • When $A = 5$, $5 \mod 4 = 1$.
   • When $A = 6$, $6 \mod 4 = 2$.
   • When $A = 7$, $7 \mod 4 = 3$.
   • When $A = 8$, $8 \mod 4 = 0$.
   • When $A = 9$, $9 \mod 4 = 1$.

   From this, $A = 3$ or $A = 7$ satisfy the condition $A \mod 4 = 3$.

4. Verify with the options: Among the given options $A = 1, 5, 7, 9$, only $A = 7$ satisfies the condition.

Final Answer:

C. 7

Would you like a further breakdown of this solution?

5 Related Questions:

1. What is the divisibility rule for other numbers like 3, 5, or 9?
2. Can you solve similar problems with different remainders and digits?
3. What happens if the number is divisible by 4 with no remainder?
4. How can modular arithmetic be used in other mathematical problems?
5. What is the fastest way to check divisibility by 4 for very large numbers?

Tip:

For divisibility problems involving remainders, focus on the specific rule and simplify the problem to just the smallest part of the number (e.g., last two digits for 4).