1. Find the smallest natural number $N$ such that $N+2$ is divisible (without remainder) by 2, $N+3$ by 3, ..., $N+10$ by 10.

ANSWER: 2520.
Solution: Note that $N$ must be divisible by $2,3,4, \ldots, 10$, therefore, N= LCM $(2,3,4, . ., 10)=2^{3} \times 3^{2} \times 5 \times 7=2520$.