Step 1 : Find prime factorization of each number.

$$\begin{aligned}1 =& 1~~~ \color{gray}{\text{ (1 is prime)}} \\[8pt]11 =& 11\\[8pt]111 =& 3\cdot37\\[8pt]1111 =& 11\cdot101\\[8pt]11111 =& 41\cdot271\\[8pt]111111 =& 3\cdot7\cdot11\cdot13\cdot37\\[8pt]1111111 =& 239\cdot4649\\[8pt]11111111 =& 11\cdot73\cdot101\cdot137\\[8pt]111111111 =& 3\cdot3\cdot37\cdot333667\\[8pt]1111111111 =& 11\cdot41\cdot271\cdot9091\\[8pt]\end{aligned}$$

(view steps on how to factor 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111 and 1111111111. )

Step 2 : Put a box around factors that are common for all numbers:

$$\begin{aligned}1 =& 1\\[8pt]11 =& 11\\[8pt]111 =& 3\cdot37\\[8pt]1111 =& 11\cdot101\\[8pt]11111 =& 41\cdot271\\[8pt]111111 =& 3\cdot7\cdot11\cdot13\cdot37\\[8pt]1111111 =& 239\cdot4649\\[8pt]11111111 =& 11\cdot73\cdot101\cdot137\\[8pt]111111111 =& 3\cdot3\cdot37\cdot333667\\[8pt]1111111111 =& 11\cdot41\cdot271\cdot9091\\[8pt]\end{aligned}$$

Note that in this example numbers do not have any common factors.

Step 3 : Multiply the boxed numbers together:

Since there is no boxed numbers we conclude that $~\color{blue}{ \text{GCD = 1} } $.

GCD Calculator