M is the largest 4 digit number, which when
divided by 4, 5, 6 and 7 leaves remainder as 2, 3, 4,
and 5 respectively. What will be the remainder
when M is divided by 9?

It will leave a remainder 1,
The numbers and the remainders all have a common difference of 2,(4-2),(5-3),(6-4),(7-5)
So, the largest 4 digit number must be of the form, N=(LCM(4,5,6,7)*k)-2,
N=420k-2, largest 4 digit digit multiple of 420 is 9660, -2 is 9658. This divided by 9 leaves a remainder of 1.