We can factor this difference as
22220038^n * (22220038^d - 1) where d = m - n > 0
For the difference to end in 8 zeros it must be a multiple of 10^8 and so must have at least 8 2's (and 8 5's) among its factors. Now, 22220038 is divisible by 2 but not by 4 (it's 11110019 * 2), so the first term contributes at most n 2's. The second term is odd and so contributes none.
For the overall to have 8 factors of 2, then, it must be that n >= 8, or that n > 7.
Posted by Paul
on 2020-10-26 09:28:48