Find the Missing Digit: Making 3_52 Divisible by 3

Q: Why does the divisibility rule for 3 work with digit sums?

This works because of how our base-10 number system relates to multiples of 3. When you add digits, you're essentially finding the remainder when the number is divided by 3!

Q: What if there are multiple digits that work?

For single-digit problems like this, there's usually only one correct answer within the range 0-9. Calculate each possibility systematically to find which digit makes the sum divisible by 3.

Q: Do I need to check every digit from 0 to 9?

Not necessarily! Since $ 10 + x $ must be divisible by 3, you can use mental math: 12 is the next multiple of 3 after 10, so $ x = 2 $.

Q: How can I quickly check if a sum is divisible by 3?

Look for sums that are multiples of 3: 3, 6, 9, 12, 15, 18, etc. You can also add the digits of the sum repeatedly until you get a single digit - if it's 3, 6, or 9, the original sum is divisible by 3!

Q: Will this method work for larger numbers with multiple missing digits?

Yes! The same principle applies - add all known digits, then determine what the missing digits must sum to in order to make the total divisible by 3. Just remember there might be multiple valid combinations.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the missing digit so that the number is divisible by 3

00:03 A number divisible by 3 is a number whose sum of digits is divisible by 3

00:10 Let's substitute the possibilities, sum them up, and check if it's divisible

01:45 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

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

$3_-52$

2

Step-by-step solution

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 $\boxed{2}$ .

3

Final Answer

$2$

Key Points to Remember

Essential concepts to master this topic

Rule: A number is divisible by 3 if digit sum is divisible by 3
Technique: Add known digits: $3 + 5 + 2 = 10$ , then find digit making sum divisible by 3
Check: With digit 2, sum is $10 + 2 = 12$ , and $12 \div 3 = 4$ ✓

Common Mistakes

Avoid these frequent errors

Checking divisibility of the whole number instead of digit sum
Don't try dividing 3252 by 3 directly = unnecessarily complex calculation! The divisibility rule for 3 works by checking if the sum of digits is divisible by 3, not the entire number. Always add up all digits first, then check if that sum divides evenly by 3.

Practice Quiz

Test your knowledge with interactive questions

Determine if the following number is divisible by 3:

$$ 352 $$

FAQ

Everything you need to know about this question

Why does the divisibility rule for 3 work with digit sums?

+

This works because of how our base-10 number system relates to multiples of 3. When you add digits, you're essentially finding the remainder when the number is divided by 3!

What if there are multiple digits that work?

+

For single-digit problems like this, there's usually only one correct answer within the range 0-9. Calculate each possibility systematically to find which digit makes the sum divisible by 3.

Do I need to check every digit from 0 to 9?

+

Not necessarily! Since $10 + x$ must be divisible by 3, you can use mental math: 12 is the next multiple of 3 after 10, so $x = 2$ .

How can I quickly check if a sum is divisible by 3?

+

Look for sums that are multiples of 3: 3, 6, 9, 12, 15, 18, etc. You can also add the digits of the sum repeatedly until you get a single digit - if it's 3, 6, or 9, the original sum is divisible by 3!

Will this method work for larger numbers with multiple missing digits?

+

Yes! The same principle applies - add all known digits, then determine what the missing digits must sum to in order to make the total divisible by 3. Just remember there might be multiple valid combinations.

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