n - first number
n + 18 - second number
[tex]n(n+18)=403\\\\n^2+18n=403\\\\(n^2+18n+81)-81=403\\\\(n+9)^2-81-403=0\\\\
(n+9)^2-484=0\\\\(n+9)^2-22^2=0\\\\(n+9-22)(n+9+22)=0\\\\(n-13)(n+31)=0\\\\\\n-13=0\qquad\vee\qquad n+31=0\\\\\boxed{n=13\in\mathbb{N}}\qquad\vee\qquad n=-31\not\in\mathbb{N}[/tex]
The numbers are 13 and 13+18 = 31
