Let's solve the problem step-by-step:

Given Problem:

The five-digit number $378A2$ has distinct digits, and it is stated that this number is divisible by 4. We are tasked to find the sum of all possible values that $A$ can take.

Step 1: Divisibility Rule for 4

A number is divisible by 4 if the number formed by its last two digits is divisible by 4. Here, the last two digits are $A2$. Therefore, $A2$ (interpreted as $10A + 2$) must be divisible by 4.

Step 2: Testing Values of $A$

To find values of $A$, we check which values make $10A + 2$ divisible by 4: $10A + 2 \mod 4 = 0 \implies 10A \mod 4 = 2 \mod 4$ Since $10 \mod 4 = 2$, the equation becomes: $2A \mod 4 = 2$ Divide through by 2: $A \mod 2 = 1$ Thus, $A$ must be odd.

The odd digits are $\{1, 3, 5, 7, 9\}$.

Step 3: Ensuring Digits are Distinct

The digits of the number $378A2$ must all be distinct. Since the digits $3, 7, 8, 2$ are already used, $A$ cannot be $3$ or $7$. This leaves possible values for $A$ as $\{1, 5, 9\}$.

Step 4: Verifying Each $A$

We check each possible value of $A$ to ensure $10A + 2$ is divisible by 4:

• For $A = 1$: $10(1) + 2 = 12$, which is divisible by 4.
• For $A = 5$: $10(5) + 2 = 52$, which is divisible by 4.
• For $A = 9$: $10(9) + 2 = 92$, which is divisible by 4.

Thus, $A$ can take the values $1, 5, 9$.

Step 5: Summing Possible Values of $A$

The sum of possible values of $A$ is: $1 + 5 + 9 = 15$

Final Answer:

The sum of all possible values of $A$ is 15.

Would you like me to explain any step further or provide additional examples?

Related Questions:

1. What is the divisibility rule for 8, and how is it different from 4?
2. If $378A2$ was required to be divisible by 8, how would the problem change?
3. How would the solution change if digits were allowed to repeat?
4. What if $A$ could only take even values? What would the result be then?
5. How does modular arithmetic help in solving divisibility problems?

Tip:

For divisibility problems, always start by identifying which part of the number (e.g., last two digits) affects the divisibility based on the given rule.