What is the smallest positive value of $x$ such that $x + 5678$ results in a palindrome?


We are tasked with finding the smallest positive value of $ x $ such that $ x + 5678 $ results in a palindrome.

### Steps to solve the problem:

1. **Understanding Palindromes:**
   A palindrome is a number that reads the same forwards and backwards. For example, 12321 and 3443 are palindromes.

2. **Start with $ x = 0 $:**
   Let's first check the value $ 5678 $ itself to see if it is a palindrome:
   \[
   5678 \quad \text{(not a palindrome)}
   \]
   Since 5678 is not a palindrome, we need to increment $ x $ and check when $ 5678 + x $ becomes a palindrome.

3. **Increment $ x $ and check each result:**
   We will begin checking successive values of $ 5678 + x $ for $ x = 1, 2, 3, \dots $ to find the first palindrome.

\[
5679 \quad \text{(not a palindrome)}
\]
\[
5680 \quad \text{(not a palindrome)}
\]
\[
5681 \quad \text{(not a palindrome)}
\]
\[
\dots
\]
\[
6006 \quad \text{(palindrome)}
\]

The smallest palindrome we encounter is $ 6006 $.

4. **Find the corresponding $ x $:**
   The value of $ x $ is the difference between 6006 and 5678:
   \[
   x = 6006 - 5678 = 328
   \]

Thus, the smallest positive value of $ x $ such that $ x + 5678 $ results in a palindrome is $ \boxed{328} $.

Would you like further details on any step? Here are five questions that extend this idea:

1. How do you define a palindrome in different bases?
2. Could the answer differ in another number system?
3. How can you find the next palindrome after a given number?
4. Are there any patterns in the differences between consecutive palindromes?
5. What are some efficient algorithms for finding the next palindrome?

**Tip:** To check whether a number is a palindrome, reverse its digits and compare it to the original number.

What is the smallest positive value of $x$ such that $x + 5678$ results in a palindrome?

Discover how to find the smallest positive value of x such that x + 5678 results in a palindrome. This problem introduces concepts of number theory and palindromes, suitable for students in Grades 6-8.

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