How many positive integers less than or equal to 6*7*8*9 solve the system of congruences:

m = 5 (mod 6)

m = 4 (mod 7)

m = 3 (mod 8)

18 I think

HCF(6,7,8) = 2

6*7*8/2 = 168 so it will be mod 168

3*7*8 * 2*9 = 6*7*8*9

3*7*8 * 18 = 6*7*8*9

So i think there will be 18.

It would be good if you come back and comment on my answer.

I've never studied modular arithmetic formally and I am guessing my way a bit.