Five times the larger of two consecutive odd integers is equal to one more than eight times the smaller. Find the integers.

Let the odd numbers be $n$ and $n+2$. Then 
\[5(n+2)=8n+1.\]
\[\Rightarrow   5n+10=8n+1    \Rightarrow 10-1=8n-5n\]
\[\Rightarrow      9=3n      \Rightarrow     n=3.\]
So the odd nombers are $n=3$ and $n+2=5$.