Find The Missing Digit: Divisible By 3 & 5!

Hey guys! Let's dive into a fun math problem today. We have a four-digit number, 47*, and we need to figure out what that missing digit (*) is. The catch? This number has to be perfectly divisible by both 3 and 5, and it needs to be an odd number. Sounds like a puzzle, right? Let's break it down step by step.

Understanding Divisibility Rules

First, let’s refresh our memory on divisibility rules. These rules are super handy shortcuts that help us determine if a number can be divided evenly by another number without actually doing long division. They're like magic tricks for math!

• Divisibility by 5: A number is divisible by 5 if its last digit is either 0 or 5. Simple enough, right?
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. For example, 123 is divisible by 3 because 1 + 2 + 3 = 6, and 6 is divisible by 3.
• Odd Number: An odd number is any whole number that cannot be divided exactly by 2. Odd numbers always end in 1, 3, 5, 7, or 9.

These divisibility rules are essential for solving our problem quickly and efficiently. Trust me, knowing these will make your life so much easier when you encounter similar questions. Think of them as your secret weapons in the world of numbers!

Applying the Divisibility Rules to Our Problem

Okay, now that we've got our divisibility rule cheat sheet ready, let's tackle our four-digit number, 47*. We know a few crucial things about this number:

1. It's divisible by both 3 and 5.
2. It's an odd number.

Let's start with the easiest rule: divisibility by 5. For 47* to be divisible by 5, the last digit (*) must be either 0 or 5. But wait! We also know the number has to be odd. This means the missing digit can't be 0 because that would make the number even. So, we've narrowed it down significantly: the missing digit must be 5. Now our number looks like 47_5_.

See how powerful those divisibility rules are? By applying just one rule, we've already eliminated one possibility and gotten much closer to our answer. This is the beauty of problem-solving in math – it's like detective work, and each rule is a clue!

Now, let's use the divisibility rule for 3. Remember, a number is divisible by 3 if the sum of its digits is divisible by 3. So, we need to add the digits of 47_5_ and see what we get: 4 + 7 + 5 = 16.

But here's the catch: 16 isn't divisible by 3. That means we need to figure out what digit we can put in the blank space to make the total sum divisible by 3. This is where a little bit of trial and error (or some clever thinking) comes in handy.

Finding the Missing Digit

We know the sum of the known digits (4, 7, and 5) is 16. To make the entire number divisible by 3, we need to add a digit to 16 that results in a multiple of 3. Let's think about multiples of 3 that are greater than 16: 18, 21, 24, and so on.

• To get to 18, we need to add 2 (16 + 2 = 18). So, one possibility is that the missing digit is 2. This would give us the number 4275.
• To get to 21, we need to add 5 (16 + 5 = 21). So, another possibility is that the missing digit is 5. This would give us the number 4575.
• To get to 24, we need to add 8 (16 + 8 = 24). So, a third possibility is that the missing digit is 8. This would give us the number 4875.

So, we have three potential numbers: 4275, 4575, and 4875. But remember, we figured out earlier that the last digit must be 5 for the number to be divisible by 5. It seems we made a small error in our logic earlier! We were focusing on the units digit, but the asterisk (*) represents the hundreds digit in this problem.

Let's backtrack and focus on the fact that the number is of the form 47*5. We already know the last digit is 5 because the number is divisible by 5. Now we need to find the missing digit in the hundreds place that makes the number divisible by 3.

So, we have the digits 4, 7, *, and 5. Let's add the known digits: 4 + 7 + 5 = 16. Again, we need to find a digit to add to 16 to get a multiple of 3. The possibilities are the same as before: adding 2, 5, or 8.

• If we add 2, we get 18, which is divisible by 3. So, 4725 is a possibility.
• If we add 5, we get 21, which is divisible by 3. So, 4755 is a possibility.
• If we add 8, we get 24, which is divisible by 3. So, 4785 is a possibility.

But there's one more crucial piece of information we haven't fully used yet: the number is odd. This might seem obvious since it ends in 5, but it helps confirm our answer. All three of our potential numbers (4725, 4755, and 4785) are odd, so this condition doesn't help us narrow it down further in this case.

However, the question states that the number is a single number. This means there should only be one correct answer. Let's carefully re-examine our steps.

We know the number is of the form 47*5, divisible by 3 and 5, and odd. We found three possibilities for the missing digit: 2, 5, and 8. This gave us the numbers 4725, 4755, and 4785. All of these are divisible by 5 and are odd.

Let's double-check the divisibility by 3:

• 4725: 4 + 7 + 2 + 5 = 18 (divisible by 3)
• 4755: 4 + 7 + 5 + 5 = 21 (divisible by 3)
• 4785: 4 + 7 + 8 + 5 = 24 (divisible by 3)

It seems we have three valid solutions! This is a bit unexpected, as math problems usually have one clear answer. Could there be a mistake in the problem statement, or are we missing something?

Let's think outside the box for a moment. The problem states "a single number," but we've found three. This suggests that there might be an additional constraint or piece of information that we're not considering. Or, it's possible the problem is flawed.

In a real-world scenario, if you encountered a problem like this, it would be a good idea to double-check the original question or consult with someone else to see if they have any insights. Sometimes a fresh pair of eyes can spot something you've missed.

For the sake of this exercise, let's assume there's a typo or missing information in the problem. If we had to choose just one answer based on the information given, we could argue that all three (4725, 4755, and 4785) are valid solutions. However, it's more likely that there's a single intended answer, and we're just not seeing the full picture.

Conclusion

So, guys, we've gone on quite a journey with this four-digit number! We used divisibility rules like pros, narrowed down possibilities, and even encountered a bit of a puzzle within the puzzle. While we didn't arrive at a single definitive answer due to the potential ambiguity in the problem statement, we learned a ton along the way.

Remember, math isn't just about finding the right answer; it's about the process of problem-solving. We explored different approaches, applied our knowledge, and thought critically. That's what really matters! And hey, maybe you can use this as a conversation starter – ask your friends if they can figure out the missing digit and see what solutions they come up with.

Keep practicing, keep exploring, and most importantly, keep having fun with math! You never know what interesting puzzles you'll encounter next. Until next time, happy problem-solving!