Let

1. Find the period of the sequence formed by the last digit of $u_n$ (it is ok just to list the first few terms of the sequence and spot the pattern - you don't need to prove it).

2. Find the last digit in the number $2^{1003}$

3. Explain why there are no $n,m\in \mathbb{N}$ such that $2^n={10}^m$

For (b), notice that $1000$ is a multiple of $4$.

For (c), if two numbers are equal, they must have the same last digit. Why is this impossible in this case?