Problem Solving

There are several good ways to look at this. Algebraically, since 256 = 2^8, we can rewrite the equation:

2^98 - 256L = N
2^98 - (2^8)L = N
2^8 (2^90 - L) = N

Notice now that on the left side, we're multiplying 256 by some other integer, 2^90 - L. Clearly we can't get 1, 2, 3 or 4 if we do that, because we're multiplying an integer by 256. The only allowable value of N here is zero.

Or you could look at it this way: When we divide 2^98 by 2^8 = 256, the remainder is zero. If we divide 256L + N by 2^8 = 256, the remainder must also be zero, because 256L + N is the same number as 2^98. But 256L is clearly divisible by 256, so N must be as well, and if N is between 0 and 4 inclusive, N can only be 0.

And there are other good approaches too.