Remainders

Here's a simple explanation without too much use of modular arithmetic language.

The number $N$ can be represented as $N = k \cdot \text{lcm}(2016,2017) + 4 = 2016 \cdot 2017 \cdot k + 4$ for some non-negative integer $k$ . The remainder when $N$ is divided by $42$ is equal to remainder when $(2016 \cdot 2017 \cdot k + 4)$ is divided by $42$ . Since $42$ completely divides $2016$ , the remainder when $2016 \cdot 2017 \cdot k$ is divided by $42$ is zero and we are left with a remainder of $\boxed{4}$ .