The 1111111111111111

Any consecutive sequence of $pq$ ones (for $p$ , $q$ positive integers) can be expressed as the product $x_p * y_{p,q}$ , where $x_p = \sum_{i=0}^{p-1} 10^i$ and and $y_(p,q) = \sum_{i=0}^q 10^{i*p}$ . Here $x_p$ is the number with $p$ consecutive ones, while $y_{p,q}$ is the number with $q$ ones, each separated by $(p-1)$ zeros. (For example, $x_3 = 111$ , while $y_{3,4} = 1001001001$ .) In the problem given, $91 = 13*7$ , so the number given is the product $x_{91} = x_{13} * y_{13,7}$ or, alternatively $x_{91} = x_7*y_{7,13}$

Consider 11 as a possible factor. For all recurring 111...11 with n recurring digits its possible to see this number as the sum (111...00 + 11) for any number in this pattern that is not 111.