We need to divide 344 into groups of 48. We can do this progressively.
Split 344 into 2 groups: 172 and 172; then split each of these into 2: 86+86 and 86+86.
We can split 86 into 2: (43+43)+(43+43) and (43+43)+(43+43). We now have 8 groups. The coaches can hold 48 people each. If we put 43 people into each coach we will therefore need 8 coaches.
Could we manage with 7 coaches? 7 coaches can hold 7×48=280+56=336 people so 344-336=8 people would have no seat on a coach. So we would need another coach for them. That means we still need 8 coaches.