Prove that a 4 digit palindrome is always divisible by 11. (2024)

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Prove that a 4 digit palindrome is always divisible by 11.? ›

1001, 1111, 1221, 1331, etc. (number starting from one should always end with one). 2002, 2112, 2222, 2332, etc. Hence each 4 digit palindrome number

All palindromes with an even number of digits are divisible by 11.

Expert-Verified Answer

The smallest 4-digit number divisible by 11 is 1001; the largest is 9999. There are therefore 819 altogether - the first one plus 1/11 of the remaining 8998.

Therefore, any 4-digit palindrome is divisible by 11.

Every number (in decimal) that is a palindrome and has an even number of digits is divisible by 11. The condition is not true the other way around however; a number times 11 will not nessecarily be an even-digit palindrome. But every palindrome with an even number of digits is a multiple of 11.

1001, 1111, 1221, 1331, etc. (number starting from one should always end with one). 2002, 2112, 2222, 2332, etc. Hence each 4 digit palindrome number is divisible by 11 as each number can be expressed as multiple of 11.

To check if a larger number is divisible by 11, find the difference between the sum of the digits at the odd places and the sum of the digits at the even places and check if it is 0 or is a multiple of 11. If yes, then the number is divisible by 11.

Divisibility rule of 11: This particular rule states that the given number can only be completely divided by 11 if the difference of the sum of digits at odd position and sum of digits at even position in a number is 0 or 11.

There are likewise 90 palindromic numbers with four digits (again, 9 choices for the first digit multiplied by ten choices for the second digit.

There are 36, 4-digit palindromic numbers, in the range of — 9999 to — 1000 AND + 1000 to + 9999 inclusive, that are divisible by 7.

Is a 5 digit palindrome number divisible by 11? ›

11011, 12221, 13431, 14641, 15851, all of which are 5-digit palindromes which are also multiples of 11.

The first few palindromic numbers are therefore are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, ...

Since 11 is a factor of both 1001 and 110, we conclude that all four digit palindromes are divisible by 11.

Divisibility Rule for 11. A number is divisible by 11 if the difference between the sum of its digits in odd places and the sum of the digits in even places is either 0 or a multiple of 11.

11 divides 2a-2b+c. 11011, 12221, 13431, 14641, 15851, all of which are 5-digit palindromes which are also multiples of 11.

A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16461) that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis.

The first few palindromic numbers are therefore are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, ...