Four-digit palindromes are always divisible by 11 because of the way our decimal number system works and a divisibility rule for 11. Let's break down why.
Understanding Four-Digit Palindromes
A four-digit palindrome has the form ABBA, where A and B are digits (0-9), and A is not zero (since it's a four-digit number). We can express this number algebraically:
ABBA = 1000A + 100B + 10B + A = 1001A + 110B
Divisibility Rule of 11
A number is divisible by 11 if the difference between the sum of its digits at odd places and the sum of its digits at even places is either 0 or divisible by 11. Applying this to ABBA:
- Sum of digits at odd places: A + B
- Sum of digits at even places: B + A
- Difference: (A + B) - (B + A) = 0
Since the difference is 0, ABBA is divisible by 11 according to the divisibility rule.
Algebraic Proof
We previously established that ABBA = 1001A + 110B. We can factor out 11:
1001A + 110B = 11(91A + 10B)
Since (91A + 10B) is an integer, 11(91A + 10B) is a multiple of 11. Therefore, any number in the form ABBA is divisible by 11.
Examples
- 2112: 2112 / 11 = 192
- 9009: 9009 / 11 = 819
- 5665: 5665 / 11 = 515
- 1331: 1331 / 11 = 121
These examples demonstrate that four-digit palindromes are indeed divisible by 11. The palindrome can always be expressed as a multiple of 11.
