10. What is the smallest 3-digit positive integer $N$ such that $2^{N}+1$ is a multiple of 5 ?
SolUtion
102

The powers of 2 (namely $2,4,8,16,32,64, \ldots$ ) have units digits which follow a sequence $2,4,8,6, \ldots$ which repeat every four terms. We may calculate that $2^{100}, 2^{101}$ and $2^{102}$ have units digits of 6,2 and 4 respectively. Therefore the first 3-digit power of 2 which is one less than a multiple of 5 is $2^{102}$.