Here is an alternative approach. Suppose k is 1792. Convert 1792 into 2^8 * 7^1. It is clear that n needs to be at least 7 in order to cover the 7, but what is the n required to provide eight 2s? Let's count: 2 provides one 2. 4 provides two 2s. 6 provides one 2. 8 provides three 2s. Now we have (1+2+1+3 = 7) 2s. We still need one more. 10 provides one 2. Now we have eight 2s. This means n==10 is the max n needed. The other factor found earlier is n==7 but 7 is less than 10. So, the answer is 10.

Edit: Just realized OP was referring to a particular submission.