To begin with, the question should read "find the 44^{th} digit".
Any number of the form abcabc is a multiple of 1001. 1001 is 7 * 11 * 13. So, any number of the form abcabc is a multiple of 13.
So, a number comprising 42 2's would be a multiple of 13, so would a number comprising 36 2's. So, in effect, we are left with a two digit number 2a, where a is the 44^{th} digit. 26 is a multiple of 13, so the 44^{th} digit should be 6.
Correct Answer: 6