Six students kept track of how much time they spent reading in a week. The...

So when we divide our numbers by how many numbers we have, we need to equal 6. Our most common number needs to be a 7.
So our mean is six, and we've got six people. This means when we add them all up, we need $6\times 6=36$ total hours.
Lets say we've got three sevens, as this is our mode.
$36-(7\times 3)=15$
We need three numbers that add up to 15, but as 7 is our mode, we cannot have three fives.
Therefore, we can use 6,4,5.
Check:
$\frac{7+7+7+6+5+4}{6}=6$