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