This is another problem I don't know how to do in a contest. Can anyone give me an hint or show me how to solve the problem? What is the trick here?
If 738 consecutive integers are added together, where the 178th number in the sequence is 4,256,815, what is the remainder when this sum is divided by 6?
The sum of the $6$ consecutive integers $n,n+1,\ldots,n+5$ is $6n+15$, which leaves a remainder of $3$ when divided by $6$, so the sum of $12$ consecutive integers is divisible by $6$. $738=12\cdot 61+6$, so we can divide the $738$ consecutive integers into $61$ blocks of $12$ consecutive integers and a block of $6$ consecutive integers. The $61$ blocks of $12$ consecutive integers have sums divisible by $6$, and the remaining block leaves a remainder of $3$, so the whole sum has a remainder of $3$.