Statement 1: If units digit of 2m is 4, then the units digit of m can be either 2 or 7. Take 62, 2*62=124. Take 67, 2*67=134. Eliminate A and D.
Statement 2: If units digit of m equals to tens digit of 2m, then m can either be 62, 2*62=124 or 74, 2*74=148. Ofcourse these are just 2 such examples. Eliminate B.
Now, m is either of 52, 57, 62, 67, 72, 77, 82, 87, 92, and 97. Of these when you do 2m, you get 104, 114, 124, 134, 144, 154, 164, 174, 184, and 194. We see that 62 and 87 meet both the conditions, hence answer is E.