Paildrome Numbers / Calculations

Palindromes are very special kind of numbers.  A palindrome can be described as a number, word, sentence, etc. which reads same forward and backward. Specifically with regards to numbers, Palindromes are numbers which are symmetrical, i.e. they remain the same even when their digits are reversed. For example 14641 is a Palindrome. In fact all the single digit numbers and numbers with same digit repeated are palindromes. So all numbers like 1,2,3…8,9,11,22,99,111,etc. are palindromes.

Features of Palindrome Numbers

1)      Reverse a non-palindromic number and add it to the original number. We will get a palindromic number by repeating this process. We may even get a palindromic number in first go. For example, let the original number be 37 (non-palindromic). Add reverse of it 73 to 37, we get 110 (not a palindromic number). Therefore repeat the process. 110 + 011 = 121 (palindromic number). Another example, 16+61 = 77 (palindromic number).

Any number that never becomes palindromic in this way is known as Lychrel Number.

2)      A palindromic number in one base may or may not be palindromic in any other base.  For example, 1991 is palindromic in both decimal and hexadecimal (7C7)

3)      Certain powers of palindromes made up of digit 1,2 and at times 3 are mostly palindromes. For example,

11^2 = 121

22^2 = 484

101*101=10201
111*111=12321 121*121=14641
202*202=40804
212*212=44944

There are, however, an infinite number of cases as demonstrated here:

11^2 = 121, 101^2 = 10201, 1001^2 = 1002001, 10001^2 = 100020001, etc.

22^2 = 484, 202^2 = 40804, 2002^2 = 4008004, 20002^2 = 400080004, etc.

4)      All even digit palindromes are divisible by 11. There are many prime palindrome numbers also like 101, 131, 151, 181, and 191

1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321