Question Number 100733 by john santu last updated on 28/Jun/20

A positive integer such as 4334 is a palindrome if it reads the same forwards or backwards. What is the only prime palindrome with an even number of digits?

Commented byRasheed.Sindhi last updated on 28/Jun/20

11 Any other palindrome must be divisible by 11 and hence composite.

Commented by1549442205 last updated on 28/Jun/20

Thank you,sir.It is correct perfectly.

Commented byRasheed.Sindhi last updated on 28/Jun/20

1331=11^3 So it′s not prime.

Commented by1549442205 last updated on 07/Jul/20

We prove the general clause following: εa arbirtary palindrome number is always divisible by 11ε.Indeedly,a palindrome is always expressed in form: A=a_1 a_2 ...a_n a_n a_(n−1) .....a_1 =^(−) a_1 a_2 ...a_n ^(−) .10^n +a_n a_(n−1) ...a_1 ^(−) =a_1 (10^(2n−1) +1)+10a_2 (10^(2n−3) +1)+... +10^(n−2) a_(n−1) .(10^3 +1)+10^(n−1) a_n (10+1) This number is divisible by 11 because 10^(2k+1) +1=(10+1)(10^(2k) −10^(2k−1) +10^(2k−2) −....+1)⋮11

Commented byjohn santu last updated on 28/Jun/20

yes..right

Commented byjohn santu last updated on 28/Jun/20

only 11