Divisors of One More than Power of 10/Number of Zero Digits Even

Theorem
Let $N$ be a natural number of the form:
 * $N = 1 \underbrace {000 \ldots 0}_{\text {$2 k$ $0$'s} } 1$

that is, where the number of zero digits between the two $1$ digits is even.

Then $N$ can be expressed as:
 * $N = 11 \times \underbrace {9090 \ldots 90}_{\text {$k - 1$ $90$'s} } 91$

Proof
By definition, $N$ can be expressed as:
 * $N = 10^{2 k + 1} + 1$

Let $a := 10$.

Then we have:

We have that:

Also see

 * Henry Ernest Dudeney: Modern Puzzles: $62$ -- Factorizing