A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is _________.
Show Hint
For any palindrome with an even number of digits, the alternating sum of digits is always zero, so it is automatically divisible by 11.
Step 1: Understanding the Concept:
A six-digit palindrome has the form $abccba$. Divisibility by 55 implies the number must be divisible by both 5 and 11. Step 2: Detailed Explanation:
Let the six-digit palindrome be $N = abccba = 100001a + 10010b + 1100c$.
1. Divisibility by 5: The last digit $a$ must be 0 or 5. Since $N$ is a six-digit number, $a \neq 0$, so $a = 5$.
The number is $5bccb5$.
2. Divisibility by 11: The difference between the sum of digits at odd places and even places must be a multiple of 11.
Odd places sum: $5 + c + b$.
Even places sum: $b + c + 5$.
Difference: $(5 + c + b) - (b + c + 5) = 0$.
Since 0 is a multiple of 11, any values for $b$ and $c$ (from $\{0, 1, ..., 9\}$) will result in a number divisible by 11.
$b$ can take 10 values ($0, 1, 2, ..., 9$).
$c$ can take 10 values ($0, 1, 2, ..., 9$).
Total such palindromes $= 1 \times 10 \times 10 = 100$. Step 3: Final Answer:
The number of such palindromes is 100.
