1. Multiplying by N causes the leading digits to increase by at least 0.5 and at most 1.0 2. Multiplying by M causes the leading digits to increase by at most 1.0 3. M - N is more than twice as large as the target number. 4. M > N and M & N both have the same number of digits.
We start at N!, then checking N+1! etc, increasing the leading digits by at least 0.5 and at most 1.0. We do this more than 2K times, so we're guaranteed to hit exactly K.
The first two conditions are pretty simple to achieve - if we're targeting 2019, then 10,003 works for N and 10,004 works for M, and we can easily find some number that fills our need for whatever target we get. For the third condition, we can multiply both N and M by 10. This increases M-N by a factor of ten, but has the exact same effect for the leading digit multiplication.