Consider the following conjecture:

If $N$ is a positive integer that consists of the digit $1$ followed by an odd number of $0$ digits and then a final digit $1$, then $N$ is a prime number.

Here are three numbers:

I $N=101$ (which is a prime number)

II $N=1001$ (which equals $7\times 11\times 13$)

III $N=10001$ (which equals $73\times 137$)

Which of these provide(s) a counterexample to the conjecture?