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# Prove that a 4−digit palindrome is always divisible by 11

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Solution

## A palindromic number is a number that remains the same when its digits are reversed. For example 9,11,33,101,141,676,1001,1771 etc. are palindrome numbers. The palindrome number with four digits takes the form abba.Here, the first digit can be chosen in 9 ways (1,2,3,4,5,6,7,8,9) and second in 10 ways (0,1,2,3,4,5,6,7,8,9) and the other two digits can be chosen on the basis of first two digits.So total number of four digit palindromic numbers are 90 and these are given as:1001,1111,1221,1331,1441,1551,1661,1771,1881,1991,.........,9009, 9119,9229,9339,9449,9559,9669,9779,9889,9999These numbers can be written as:1001=990+11=11(90+1)1111=1100+11=11(100+1)1221=1210+11=11(110+1) and so on.....Hence, each 4 digit palindrome number is divisible by 11.

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