Question

a palindrome is a number that is the same when read forwards and backwards, such as $43234$. what is the smallest five-digit palindrome that is divisible by $11$?

Answer 1

To find the smallest five-digit palindrome divisible by 11, we need to consider the form of the palindrome and the divisibility rule of 11. By testing different values for the digits, we can find that the smallest palindrome is 10101.

Step-by-step explanation:

To find the smallest five-digit palindrome that is divisible by 11, we can start by noting that the palindrome is an odd number since the first and last digits must be the same. The second digit can be any number since it doesn't affect divisibility by 11. Therefore, we can consider the palindrome to be in the form $ababa$, where $a$ and $b$ are digits.

Since the palindrome is divisible by 11, the difference between the sum of the digits in odd positions (a + a + a) and the sum of the digits in even positions (b + b) must be divisible by 11. In this case, the difference is 2a - 2b = 2(a-b), which must be divisible by 11.

Through trial and error, we can test different values for $a$ and $b$ until we find the smallest five-digit palindrome that is divisible by 11. Starting with $a=1$ and $b=1$, we find that the palindrome 10101 is divisible by 11. Therefore, the smallest five-digit palindrome divisible by 11 is 10101.

Answer 2

the answer is 10901

Step-by-step explanation:

A palindrome is a number that reads the same way forwards and backward(such as 43234). A multiple of 11 has an 'alternating sum' that is a multiple of 11(such as 121 since 1+1-2=0 which is a multiple of 11). The smallest 5-digit palindrome that is a multiple of 11 is 10901 since it reads the same way forwards and backward. 1+9+1-0-0=11 which is a multiple of 11.