Find the largest integer less than 8000

Question

Is there a simple way to solve the problem?

Find the largest integer less than 8000 which leaves a remainder of 1 when divided by 2, 3, 4, 5, 6, 7, 8, and 9.

Answer 1

Review the Chinese remainder theorem. Clearly $1$ leaves the proper remainder. What is the next number that does?

Answer 2

Suppose $N$ leaves remainder $1$ when divided by $2,3,4,5,6,7$. This happens if and only if $2,3,4,5,6,7$ all divide $N-1$, which happens if and only if their least common multiple, $420$ divides $N-1$.

Therefore $N=420k+1$. It is easy to check the largest multiple of $420$ under $8000$ is $420\times 19=7980$. So the largest value for $N$ is $7981$