Let the five consecutive integers be $n-2,$ $n-1,$ $n,$ $n+1,$ and $n+2.$
The sum of the interior angles of a pentagon is $180(5-2)=540$ degrees.
Setting up an equation, we have \[
(n-2)+(n-1)+n+(n+1)+(n+2)=540,
\] which simplifies to $5n=540$ or $n=108.$
The largest angle is $n+2=108+2=\boxed{110}$ degrees.
The answer is: 110
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
The number of degrees in the measures of the interior angles of a convex pentagon are five consecutive integers. What is the number of degrees in the largest angle?

### Answer:
