What is the largest integer that is a solution of $13x + 8 < 35$?
Subtracting $8$ from both sides of the inequality, we have $$13x < 27.$$ Dividing both sides by $13$ gives $$x < \frac{27}{13}.$$ Writing this in terms of mixed numbers, we have $$x < 2\frac1{13}.$$ The largest integer satisfying this inequality is $x=\boxed{2}$.