Find the Missing Digit: Making 3_52 Divisible by 3

Complete the number so that it is divisible by 3 without a remainder:

$3_-52$

To solve this problem, we'll follow these steps:

• Step 1: Add the known digits of the number.
• Step 2: Determine which missing digit results in a total sum divisible by 3.
• Step 3: Validate the result against given choices.

Now, let's execute these steps:

Step 1: Calculate the sum of the known digits.

The given number format is $3\_52$. The known digits are 3, 5, and 2. Calculate their sum: $3 + 5 + 2 = 10$.

Step 2: Determine the missing digit that completes the sum to be divisible by 3.

Let's denote the missing digit as $x$. The total sum of the digits is $10 + x$. For the number to be divisible by 3, $10 + x$ must also be divisible by 3.

Examine the possible values for $x$:

• If $x = 0$, then $10 + 0 = 10$ (not divisible by 3).
• If $x = 1$, then $10 + 1 = 11$ (not divisible by 3).
• If $x = 2$, then $10 + 2 = 12$ (divisible by 3).
• If $x = 3$, then $10 + 3 = 13$ (not divisible by 3).
• Continue in this manner for $x = 4$ to $9$.

The correct $x$ that results in a sum divisible by 3 is 2.

Step 3: Validate this result against the provided choices.

Among the choices given, $2$ is indeed an option. Thus, the correct missing digit is $2$.

Therefore, the solution to the problem is that the missing digit $x$ is $\boxed{2}$.