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 420×19=7980. So the largest value for N is 7981
Answers & Comments
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 420×19=7980. So the largest value for N is 7981