822b Divisible By 4? Find 'b' And The Numbers!

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822b Divisible by 4? Find 'b' and the Numbers!

Hey guys! Let's dive into a cool math problem. We're talking about the number 822b, where 'b' is a single digit. The kicker? We know this number is perfectly divisible by 4. Our mission, should we choose to accept it, is to figure out what values 'b' can be, and then find the smallest and largest numbers that fit the bill. Sounds fun, right? Buckle up, because we're about to explore the awesome world of divisibility rules and number theory! This problem is a great example of how a little bit of knowledge can unlock some interesting mathematical puzzles. So, let's get started. We'll break it down step by step to make sure everyone understands, regardless of your math background. It's like a fun treasure hunt, where the treasure is the answer! We will use strong and bold tags to emphasize important concepts. I think it is important to first understand the divisibility rule for 4, which is key to solving this problem. Essentially, the rule states that a number is divisible by 4 if the number formed by its last two digits is divisible by 4. So, instead of checking the whole number, we just focus on the last two digits. This makes things much easier!

(a) Finding the Possible Values of 'b'

Alright, let's get down to business and figure out what values 'b' can take. As we just discussed, the divisibility rule for 4 is our golden ticket here. We only need to focus on the last two digits of the number 822b, which are '2b'. We need to find single-digit values for 'b' (remember, 'b' is a digit, so it can be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9) that make '2b' divisible by 4.

Let's test each possible value of 'b' to see if the two-digit number '2b' is divisible by 4.

  • If b = 0: The number is 20. 20 / 4 = 5. Bingo! 20 is divisible by 4.
  • If b = 1: The number is 21. 21 / 4 = 5.25. Nope, not divisible.
  • If b = 2: The number is 22. 22 / 4 = 5.5. Still a no-go.
  • If b = 3: The number is 23. 23 / 4 = 5.75. Not divisible.
  • If b = 4: The number is 24. 24 / 4 = 6. Yes! 24 is divisible by 4.
  • If b = 5: The number is 25. 25 / 4 = 6.25. Nope, not divisible.
  • If b = 6: The number is 26. 26 / 4 = 6.5. Still not divisible.
  • If b = 7: The number is 27. 27 / 4 = 6.75. Not divisible.
  • If b = 8: The number is 28. 28 / 4 = 7. Absolutely! 28 is divisible by 4.
  • If b = 9: The number is 29. 29 / 4 = 7.25. Nope, not divisible.

So, after all that testing, we've found that 'b' can be 0, 4, or 8. Therefore, the values of b that make 822b divisible by 4 are 0, 4, and 8. Using bold tags to emphasize these final values is extremely important. This is one of the most important results that we need to find! Great job, everyone! We've successfully cracked the code and discovered all the possible values for 'b'. This is a pretty straightforward process, but it highlights the importance of divisibility rules. Now that we've nailed down the possible values for 'b', we can move on to the next part of the problem.

We successfully completed the first part of the problem. This is a crucial step because it sets the stage for finding the smallest and largest possible numbers that fit our criteria. We are making great progress! We can move on to the next part and find the smallest and largest possible numbers. The first part is easier and the second one needs a bit of thinking, but let's go!

(b) Finding the Smallest and Largest Numbers

Now that we know 'b' can be 0, 4, or 8, let's find the smallest and largest possible numbers that fit the form 822b. This part is pretty easy, but let's take a look. We know our number looks like 822b. To find the smallest number, we need to use the smallest possible value for 'b', which we found to be 0. So, the smallest number is 8220.

To find the largest number, we use the largest possible value for 'b', which is 8. So, the largest number is 8228.

Therefore, the smallest number is 8220, and the largest number is 8228. Congratulations, guys! We've reached the finish line and solved the entire problem! We identified the possible values for 'b', and then we easily figured out the smallest and largest numbers. This is a great example of how understanding divisibility rules can help you solve number theory problems.

We successfully found the smallest and largest numbers that are divisible by 4, and we've completed the entire problem! It's important to remember that this whole process hinges on understanding the divisibility rule of 4. So, the next time you encounter a problem like this, remember this method. Don't worry if you didn't get it right away. Practice makes perfect, and with a little bit of effort, you'll be able to solve these kinds of problems with ease. This problem is a classic example of how number theory works in action. Keep practicing and keep learning, and you'll be amazed at how much you can accomplish. The more you work with numbers, the more intuitive and natural the concepts will become. And always remember to have fun with it! Keep practicing! In the world of mathematics, a strong foundation in divisibility rules is super helpful.

Summary of Results

Let's summarize our findings to make sure everything is crystal clear.

  • Possible values of 'b': 0, 4, 8
  • Smallest number: 8220
  • Largest number: 8228

We hope this explanation was helpful and easy to follow. Remember, the key takeaway is the divisibility rule for 4 and how to apply it. Keep up the great work, and don't be afraid to tackle more math problems. You got this! This problem is a fantastic example of a simple math puzzle that reinforces important concepts. This type of problem is very useful to practice your math skills. It also provides a great foundation for more complex mathematical concepts later on.

I hope that this explanation has been clear and helpful! Keep practicing your math skills, and don't hesitate to ask questions if you get stuck. You're all doing great, and with a little bit of effort, you'll continue to grow and develop your mathematical abilities. Number theory is a fascinating field, and there's a lot more to explore. Keep up the good work, and always remember to have fun with it! Understanding divisibility rules is crucial for tackling various number theory problems. You're on your way to becoming math whizzes! Keep practicing and keep exploring the amazing world of mathematics!