Abox of marbles can be equally shared among 6,7,8 students with 4 marbles left over each time. what is the least possible number of marbles in the box?

Let the number of marbles be M, 

then, 

M=6a+4 (when shared among 6 students)

M=7b+4 (when shared among 7 students)

M=8a+4 (when shared among 8 students)

thus

M-4=6a

M-4=7b

M-4=8c

That is M-4 is the smallest number which is a multiple of 6, 7 and 8

Thus we need to find the LCM(6, 7, 8)

LCM(6, 7, 8)=LCM(2*3, 7, 2*2*2) = 2*2*2*3*7=8*21=168

The smallest M-4 is 168, 

so the smallest M is 168+4=172

172