Smallest Even 5-Digit Number Summing To 22

by SLV Team 43 views

Let's dive into a fun mathematical puzzle! We're on the hunt for the smallest even five-digit number, but there's a catch – all its digits must be different, and they need to add up to a total of 22. Sounds like a mission, right? Don't worry; we'll break it down step by step to make it super easy to understand. Finding the smallest even five-digit number that meets specific criteria is a fascinating challenge. This problem combines the concepts of number theory, digit manipulation, and logical reasoning. To solve this, we must strategically choose digits that not only sum to 22 but also adhere to the constraints of being distinct and forming the smallest possible even number.

Understanding the Requirements

First, let's clarify what we're looking for:

  • Five-Digit Number: It needs to be between 10,000 and 99,999.
  • Even: The last digit must be 0, 2, 4, 6, or 8.
  • Distinct Digits: No digit can be repeated.
  • Sum of Digits is 22: When you add all the digits together, they must equal 22.

Strategy for Finding the Number

To get the smallest possible number, we want to keep the leftmost digits as small as possible. Here’s how we can approach this:

  1. Start with the Leftmost Digit: The smallest possible digit for the ten-thousands place is 1. This helps keep the number as small as possible.
  2. Next Digit: For the thousands place, we want the smallest digit that hasn't been used. So, let's try 0. Our number now looks like 10,XXX.
  3. Hundreds Place: Again, we want a small digit. We can't use 0 or 1, so let's try 2. Now we have 10,2XX.
  4. Consider the Last Digit (Units Place): Since the number needs to be even, the last digit must be 0, 2, 4, 6, or 8. However, 0 and 2 are already taken, so we need to consider 4, 6, or 8.
  5. Calculate Remaining Sum: We need the digits to add up to 22. So far, we have 1 + 0 + 2 = 3. That means the remaining two digits must add up to 22 - 3 = 19.

Finding the Right Digits

Now, let’s figure out the last two digits. We know one of them has to be 4, 6, or 8 to keep the number even.

  • If the last digit is 4: The remaining digit would need to be 19 - 4 = 15. That’s not possible since we can only use single digits.
  • If the last digit is 6: The remaining digit would need to be 19 - 6 = 13. Again, not possible.
  • If the last digit is 8: The remaining digit would need to be 19 - 8 = 11. Still not possible.

Okay, it seems like our initial choices might not work. Let's backtrack a bit and adjust our strategy.

Adjusting the Strategy

Since 1, 0, and 2 didn't lead to a solution, let’s try a different approach. We'll keep 1 as the first digit, but let's increase the second digit to see if we can find a combination that works.

  1. Keep 1 as the First Digit: This is still the best way to keep the number small.
  2. Try 2 as the Second Digit: So, we have 12,XXX. This is the next smallest option.
  3. Next Digit: Let’s try 0. Now we have 12,0XX.
  4. Consider the Last Digit (Units Place): It needs to be even, so let’s consider 4, 6, or 8 (since 0 and 2 are taken).
  5. Calculate Remaining Sum: We need the digits to add up to 22. So far, we have 1 + 2 + 0 = 3. That means the remaining two digits must add up to 22 - 3 = 19.

Let's rethink this. To minimize the number, we want the smallest digits in the leftmost places. So, let's start with 1 as the first digit.

Refining the Approach

  1. First Digit: Start with 1. The number looks like 1XXXX.
  2. Second Digit: Use the smallest available digit, which is 0. The number is now 10XXX.
  3. Third Digit: Use the next smallest, which is 2. Now the number is 102XX.
  4. Fifth Digit (Units): It has to be even, so let's consider the smallest options: 4, 6, 8.
  5. The Sum Problem: The digits need to add up to 22. So far, 1 + 0 + 2 = 3. The remaining two digits must sum to 19.

If we try to end with 4, the other digit would be 15 (impossible). If we try to end with 6, the other digit would be 13 (impossible). If we try to end with 8, the other digit would be 11 (impossible).

Another Adjustment

Let’s keep 1 as the first digit and try 3 as the second digit. So we have 13XXX. Then 0, so 130XX. Now, we need two digits that add up to 22 - (1+3+0) = 18.

To keep the number small, let's try the smallest even number for the last digit. If the last digit is 2 (we can't use 0), the other digit would be 16, which is not possible.

Let's try the next smallest for the third digit, which is 4. So we have 134XX. Now, we need two digits that add up to 22 - (1+3+4) = 14.

If the last digit is 0, we can't use it. If the last digit is 2, the remaining digit would be 12, which is not possible. If the last digit is 6, the remaining digit would be 8. So we have 13486.

  • Digits: 1, 3, 4, 8, 6
  • Sum: 1 + 3 + 4 + 8 + 6 = 22
  • Even: Yes (ends in 6)
  • Distinct: Yes

The Solution

So, after all that brainstorming and adjusting, we found our number! The smallest even five-digit number with distinct digits that add up to 22 is 13,486. Isn't that awesome? Sometimes, solving math problems feels like going on an adventure!

Final Answer

The smallest even five-digit number with distinct digits that has a digit sum of 22 is 13486.