How many 100-digit palindromic numbers are prime?
A palindromic number is one which reads the same way forwards
and backward, such as 14599541.
There are no 100-digit palindromic primes; all 100-digit palindromic
numbers are divisible by 11.
To see this, first observe that 102k+1 + 1 is divisible
by 11, for k = 0, 1, 2, ..., since:
by the binomial theorem, and the quantity on the right is clearly a
multiple of 11.
Now, if N is a 100-digit palindromic number, we have:
All of the expressions in brackets are divisible by 11, so N is divisible
by 11 as well.