Number Game based on remainder theory

Question

Which is the smallest number that when divided by 5 gives 4 as remainder, when divided by 6 gives 5 as remainder, when divided by 7 gives 6 as remainder, when divided by 8 gives 7 as remainder, when divided by 9 gives 8 as remainder and when divided by 10 gives 9 as remainder?

Answer 1

We have that such number $N$ is congruent to $-1$ modulo $5,6,7,8,9,10$. Then $N+1=\mbox{lcm}(5,6,7,8,9,10)=5\cdot 7\cdot 8\cdot 9=2520$. Hence $N=2519$.