2. Problem: Find the smallest positive integer $N$ satisfying the following three properties.
- $N$ leaves a remainder of 5 when divided by 7 .
- $N$ leaves a remainder of 6 when divided by 8 .
- $N$ leaves a remainder of 7 when divided by 9 .
Solution: We notice that $\{5,6,7\}$ are each 2 less than $\{7,8,9\}$ respectively. Therefore $N+2$ must be a multiple of 7,8 and 9 . Since 7,8 and 9 are pairwise coprime, this means that
$(N+2)$ is a multiple of $7 \times 8 \times 9=504$.

Therefore the smallest positive possibility for $N+2$ is 504 . Thus $N=502$.