Finding The Missing Digit: Divisibility By 3 Explained

Hey guys! Let's dive into a cool math problem that's all about finding a missing digit in a number. We're going to use the concept of divisibility by 3. Specifically, we're looking at the three-digit number 47?, where the question mark represents a missing digit. The catch? We know this number is perfectly divisible by 3. Our mission is to figure out what values the missing digit can be, and then find the sum of those values. Sounds fun, right?

So, how do we tackle this? Well, the key lies in the divisibility rule for 3. This rule is super handy and makes this problem a breeze. Basically, a number is divisible by 3 if the sum of its digits is also divisible by 3. This means if we add up all the digits in our number, and the result is a multiple of 3 (like 3, 6, 9, 12, and so on), then the original number is also divisible by 3. This is our secret weapon here!

Let's apply this to our number, 47?. We know two of the digits: 4 and 7. The third digit is the one we're trying to find. Let's call this missing digit 'x'. According to our rule, the sum of the digits (4 + 7 + x) must be divisible by 3. That simplifies to (11 + x) must be divisible by 3. Now, we just need to find the values of 'x' that make this true. Remember, 'x' has to be a single digit, meaning it can be any number from 0 to 9. Let's start testing values!

We will find the value of x such that 4 + 7 + x is divisible by 3. We can write this as 11 + x is divisible by 3. x represents a digit, and can take values from 0 to 9.

Decoding Divisibility: Unveiling the Magic of 3

Alright, folks, let's break down this divisibility rule a bit more. It's a fundamental concept in number theory, and understanding it can make these kinds of problems much easier to solve. The core idea is that instead of doing the actual division, which can be time-consuming, we can quickly determine if a number is divisible by 3 just by looking at the sum of its digits. Isn't that neat?

Think about it this way: when you divide a number by 3, you're essentially trying to group it into sets of 3. If you can do this perfectly, with no leftovers, then the number is divisible by 3. The divisibility rule gives us a shortcut to check if this perfect grouping is possible without actually doing the division. It works because of the way our number system is structured. Each place value (ones, tens, hundreds, etc.) represents a power of 10. And, interestingly, when you divide any power of 10 by 3, the remainder is always 1. For example, 10 divided by 3 leaves a remainder of 1, 100 divided by 3 leaves a remainder of 1, and so on.

So, when we add up the digits, we're essentially finding out how many 'leftovers' we have when we divide each place value by 3. If the total number of leftovers is a multiple of 3, then the original number is also divisible by 3. This is why adding the digits works! Let's get back to the problem.

We know that the possible values of x, from 0 to 9, are those that make 11 + x divisible by 3. Let's try it out.

If x = 0, then 11 + 0 = 11. 11 is not divisible by 3.

If x = 1, then 11 + 1 = 12. 12 is divisible by 3. Therefore, 471 is divisible by 3.

If x = 2, then 11 + 2 = 13. 13 is not divisible by 3.

If x = 3, then 11 + 3 = 14. 14 is not divisible by 3.

If x = 4, then 11 + 4 = 15. 15 is divisible by 3. Therefore, 474 is divisible by 3.

If x = 5, then 11 + 5 = 16. 16 is not divisible by 3.

If x = 6, then 11 + 6 = 17. 17 is not divisible by 3.

If x = 7, then 11 + 7 = 18. 18 is divisible by 3. Therefore, 477 is divisible by 3.

If x = 8, then 11 + 8 = 19. 19 is not divisible by 3.

If x = 9, then 11 + 9 = 20. 20 is not divisible by 3.

So the possible values of the missing digit are 1, 4, and 7. Now we can find their sum. This gives us 1 + 4 + 7 = 12

Solving for 'x': Finding the Missing Digits

Now, let's roll up our sleeves and actually find the possible values for 'x'. We know that 11 + x needs to be divisible by 3. Let's start with the smallest possible value for 'x', which is 0. If x = 0, then 11 + 0 = 11. Is 11 divisible by 3? Nope. So, 0 isn't a possible value for 'x'. Next, let's try 1. If x = 1, then 11 + 1 = 12. Hey, 12 is divisible by 3! That means 1 is a valid value for our missing digit. We can confirm this by seeing that 471 / 3 = 157.

Let's keep going. If x = 2, then 11 + 2 = 13. Not divisible by 3. If x = 3, then 11 + 3 = 14. Nope. If x = 4, then 11 + 4 = 15. Bingo! 15 is divisible by 3, so 4 is another possible value. We can confirm this by seeing that 474 / 3 = 158.

Now, let's go on: if x = 5, then 11 + 5 = 16. Not divisible by 3. If x = 6, then 11 + 6 = 17. Nope. If x = 7, then 11 + 7 = 18. Awesome! 18 is divisible by 3, so 7 is a possible value for x. We can confirm this by seeing that 477 / 3 = 159. Lastly, if x = 8, then 11 + 8 = 19. Nope, and If x = 9, then 11 + 9 = 20. Not divisible by 3. Therefore, the possible values for 'x' are 1, 4, and 7.

Summing It Up: Calculating the Final Answer

Alright, we've done the hard part – figuring out the possible values for our missing digit. Now, we just need to add those values together to get our final answer. We found that the possible values for 'x' are 1, 4, and 7. So, the sum is 1 + 4 + 7, which equals 12. And there you have it! The sum of the possible values for the missing digit in the number 47? is 12.

See, that wasn't so bad, right? We used a simple divisibility rule to solve a math problem that might have seemed a bit tricky at first. This is a great example of how understanding basic math concepts can help you tackle more complex problems. Keep practicing, and you'll become a math whiz in no time!

Let's recap what we've learned:

• Divisibility Rule of 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Applying the Rule: We used this rule to find the missing digit in 47?.
• Finding the Possible Values: We determined that the missing digit could be 1, 4, or 7.
• Calculating the Sum: We added those values together (1 + 4 + 7) to get our final answer, 12.

Practical Applications of Divisibility Rules

So, why is knowing the divisibility rule for 3 important, you might ask? Well, it's not just about solving math problems in a classroom, guys. These rules have some really cool, real-world applications. They can help you with quick mental calculations, checking your work, and even understanding how computers work.

Imagine you're at the grocery store, and you want to quickly check if the total cost of your items is divisible by 3. Knowing the divisibility rule lets you do this in your head, without needing a calculator. This can be handy for budgeting and keeping track of your spending. Or, maybe you're a programmer, and you're working on an algorithm that involves dividing numbers. Understanding divisibility rules can help you optimize your code and make it more efficient. In fact, many computer algorithms rely on the principles of number theory, including divisibility rules.

Beyond practical applications, these rules also help build a stronger foundation in mathematics. They teach you to think critically, to recognize patterns, and to understand the relationships between numbers. These skills are invaluable in all areas of life, from managing your finances to understanding scientific concepts.

Final Thoughts and Next Steps

Well, guys, that wraps up our exploration of divisibility by 3 and finding the missing digit. I hope you found this explanation helpful and that you feel more confident about tackling these types of problems in the future. Remember, the key is to understand the underlying concepts and to practice, practice, practice!

If you enjoyed this, there are plenty of other divisibility rules to explore. You could check out the rules for 2, 4, 5, 6, 9, and 11. Each one has its own unique trick, and learning them can be a fun way to improve your math skills. Also, consider trying some similar problems. You can change the numbers, the divisibility rule, or the number of missing digits to create new challenges. And if you have any questions, don't hesitate to ask! Math is all about learning, exploring, and having fun, and I'm always happy to help. Keep up the great work, and keep exploring the amazing world of mathematics! Until next time, keep crunching those numbers, and happy calculating!