Smallest 4-Digit Number Divisible By 3: A Math Puzzle

by Luna Greco 54 views

Hey everyone! Let's dive into a fascinating math puzzle today. We've got three digits – 5, 3, and 6 – and our mission is to construct the smallest possible 4-digit number that's perfectly divisible by 3. Sounds intriguing, right? This isn't just about randomly throwing numbers together; we need a strategy, a bit of number sense, and a sprinkle of divisibility rules. So, buckle up, math enthusiasts, as we embark on this numerical adventure!

Understanding Divisibility Rule of 3: The Key to Our Solution

Before we start juggling the digits, let's quickly recap the divisibility rule of 3. This rule is our golden ticket to solving this puzzle efficiently. It states that a number is divisible by 3 if the sum of its digits is divisible by 3. Remember this, guys – it's crucial! For example, take the number 123. The sum of its digits (1 + 2 + 3) is 6, which is divisible by 3. Therefore, 123 is also divisible by 3. See how that works? Now, with this rule in our arsenal, we can strategically build our 4-digit number.

To make this concept crystal clear, let’s consider why this rule works. It's rooted in the fundamental properties of our number system (base-10). When we divide a power of 10 by 3, the remainder is always 1. Think about it: 10 divided by 3 leaves a remainder of 1, 100 divided by 3 leaves a remainder of 1, 1000 divided by 3 also leaves a remainder of 1, and so on. This means that each digit in a number contributes its face value to the remainder when the entire number is divided by 3. Therefore, if the sum of the digits is divisible by 3, the entire number is also divisible by 3. Understanding this underlying principle makes the divisibility rule of 3 not just a trick, but a powerful tool for number manipulation.

Now, let's apply this to our specific problem. We have the digits 5, 3, and 6. Their sum is 5 + 3 + 6 = 14. This sum isn't divisible by 3, which means any 3-digit number we make using these digits won't be divisible by 3. But we need a 4-digit number! This means we'll need to reuse one of our digits. The question then becomes: which digit should we repeat, and how do we arrange them to create the smallest possible 4-digit number that is divisible by 3? Keep that thought simmering as we move on to the next stage of our strategy.

Crafting the Smallest Number: A Digit-by-Digit Strategy

Now comes the fun part – actually building our number! Remember, we want the smallest 4-digit number possible. To achieve this, we need to think strategically about the placement of each digit. The thousands place is the most significant, so we want the smallest digit possible in that position. Looking at our options (5, 3, and 6), 3 is the clear winner for the thousands place. This gives us a head start: our number will be 3 _ _ _.

Next up is the hundreds place. Again, we want the smallest digit available. But hold on! We need to remember the divisibility rule of 3. We know that the sum of our digits needs to be divisible by 3. Currently, we have 3 + _ + _ + _ . The sum of our original digits (5, 3, and 6) was 14. Since 14 isn't divisible by 3, we need to add a digit that will make the total sum divisible by 3. The next multiple of 3 after 14 is 15, so we need to add 1. However, we don't have a '1' digit. The next multiple of 3 is 18, meaning we need to add 4 to our current sum. We don’t have ‘4’ either. The next is 21 so we need to add 7. Still don’t have it. Wait a minute! What if we repeated one of our original digits? Let's explore that.

If we repeat the 3, the sum becomes 14 + 3 = 17 (not divisible by 3). If we repeat the 5, the sum becomes 14 + 5 = 19 (not divisible by 3). But if we repeat the 6, the sum becomes 14 + 6 = 20 (still not divisible by 3…almost there). So we need to think further. We need a sum that’s divisible by 3. The closest sums are 18 and 21. If we aim for 18, we need to add 4 to 14 (and we don’t have ‘4’). If we aim for 21, we need to add 7 (and we don’t have ‘7’). Hmmm…let’s revisit our digit choices. Perhaps we missed something crucial.

Let's systematically analyze each possibility. We already know 3 goes in the thousands place. Now, for the hundreds place, let's consider our options. If we put 3 in the hundreds place, we have 33 _ _. Our digits left are 5, 6, and we need to repeat one. The current sum is 3 + 3 + 5 + 6 = 17. To get to 18, we need to add 1 (can't do). To get to 21, we need to add 4 (can't do). If we try putting 5 in the hundreds place, we get 35 _ _. The digits left are 3, 6, and a repeated one. Sum: 3 + 5 + 6 = 14. Repeating 3: 17 (no). Repeating 5: 19 (no). Repeating 6: 20 (no). Finally, let's try 6 in the hundreds place: 36 _ _. Digits left: 3, 5, and a repeated digit. Sum: 3 + 6 = 9. So the original digits 5,3, and 6 have a sum of 14. Adding this to the 9, we get a 23. Repeating 3 : 26 ( no). Repeating 5 : 28 ( no). Repeating 6 : 29 (no). OKAY! This feels like we’re going in circles! Let's take a step back and think about the smallest numbers we can make with these digits, regardless of divisibility for a moment.

The Eureka Moment: The Correct Approach

Okay, guys, sometimes the best way to solve a problem is to rethink your approach entirely. We were so focused on adding the right digit to make the sum divisible by 3, that we forgot a crucial aspect of our goal: making the smallest number. Let's get back to basics. The smallest 4-digit number will start with the smallest possible digit in the thousands place, which we correctly identified as 3. So we have 3 _ _ _.

To keep the number as small as possible, we should aim to use the smallest digits in the remaining places as well. This means we should consider repeating the smallest digit, which is 3. If we repeat 3, we get the digits 3, 3, 5, and 6. Now, let's check the sum: 3 + 3 + 5 + 6 = 17. As we've already established, 17 isn't divisible by 3. So, repeating 3 won't work.

The next smallest digit is 5. Let's try repeating 5. Our digits would be 3, 5, 5, and 6. The sum is 3 + 5 + 5 + 6 = 19. Nope, not divisible by 3 either. Let’s see, our next attempt would be to repeat 6 giving us 3,5,6 and 6. Adding these up gives us 3 + 5 + 6 + 6 = 20. Nope, not a sum divisible by three.

So what about if we repeat 6? Our digits become 5, 3,6 and 6. Their sum is 5 + 3 + 6 + 6 = 20. 20 isn’t divisible by 3. So this won’t work. It feels like we need to systematically look at possible 4 digit number combinations and test them for divisibility by 3. Is there a more efficient way?

Wait a second! If we have 3 _ _ , to make the number smallest, we need to place digits in ascending order where possible. We have 5 and 6, so the next smallest digit is 5. So we get 35 _. To make this work, let's think about the lowest number we could add to this and then see if its divisible by 3. So 3 + 5 is 8. If we use the 2 remaining digits which are both 6, we get 8 + 6 + 6 = 20. Nope not divisible by 3! Let's keep our first 2 digits as 35 and switch our remaining digits around.

It seems we’ve hit a bit of a snag, guys. We’ve tried repeating each digit, and none of the sums are divisible by 3. This means we need to adjust our strategy slightly. We can't just blindly repeat a digit; we need to be more strategic about which digit we choose to repeat and how we arrange them. This calls for a bit of trial and error, but with a clear head and our divisibility rule firmly in mind.

Let’s revisit the idea of creating the smallest number first, and then checking for divisibility. We know 3 is the smallest digit, so 3 should be in the thousands place. Then, to minimize the number, we should try to put the next smallest digit, 3, in the hundreds place. So we have 33 _ _. The remaining digits are 5 and 6. The sum of 3 + 3 + 5 + 6 = 17. We need to increase this sum to the nearest multiple of 3, which is 18. To do this, we could reduce any one of our digits and replace it with one more than the original. But to reduce the number’s size overall, we should keep this change as small as possible.

So if we change 3 + 3 + 5 + 6 = 17 to a value of 18, we need to add 1 to the sum of the digits. We can change the 5 to a 6. This gets us to our sum which will now be 18 and divisible by 3. Therefore we have 3, 3, 6 and 6! This is our Eureka moment! The smallest number should be 3366! Let’s check this by adding the numbers together: 3+3+6+6 = 18. 18/3 = 6! Great, let's try diving the number by 3 to check we get a whole number. 3366/3 = 1122. Yipee, it works!!

The Grand Finale: Assembling the Solution

Alright, after some twists and turns, we've cracked the code! We figured out that the digits we need are 3, 3, 6, and 6. Now, the final step is to arrange these digits to form the smallest possible number. Remember, we want the smallest digits in the most significant places. So, placing the two 3s first, we have 33 _ _. Then, placing the 6s gives us 3366.

So, the smallest 4-digit number divisible by 3 that can be written with the digits 5, 3, and 6 (with repetition allowed) is 3366! How cool is that? We used the divisibility rule of 3, a bit of logical thinking, and a dash of perseverance to arrive at the solution. Math puzzles like these are not just about finding the right answer; they're about honing our problem-solving skills and enjoying the journey of discovery.

Key Takeaways and Pro Tips

Before we wrap up, let's highlight some key takeaways from this numerical quest:

  • Divisibility Rules are Your Friends: Master the divisibility rules for various numbers (3, 9, 4, 8, etc.). They're powerful shortcuts that can save you time and effort.
  • Think Strategically: Don't just jump into calculations. Take a moment to plan your approach. Consider what the problem is asking and what tools you have at your disposal.
  • Smallest to Largest: When trying to form the smallest number, start by placing the smallest digits in the most significant places (thousands, hundreds, etc.).
  • Don't Be Afraid to Re-evaluate: If your initial approach isn't working, don't be afraid to step back, rethink, and try a different strategy. Sometimes, a fresh perspective is all you need.
  • Practice Makes Perfect: The more math puzzles you solve, the better you'll become at recognizing patterns, applying rules, and thinking critically.

So, there you have it, guys! We've successfully navigated this number puzzle. I hope you enjoyed this mathematical adventure as much as I did. Keep exploring, keep questioning, and keep those mental gears turning! Until next time, happy problem-solving!