Question

Aratati ca numarul n=$\sqrt{(10*x+5)*(10*x+3)+1}$ este numar natural, pentru orice cifra nenula x.

Answer 1

$\sqrt{(10x+5)(10x+3)+1} = \\ \\ = \sqrt{\big[(10x+3)+2\big](10x+3)+1}\\ \\ = \sqrt{(10x+3)^2+2(10x+3)+1}\\ \\ = \sqrt{\big[(10x+3)+1\big]^2}\\ \\ = \sqrt{(10x+4)^2}\\ \\ = |10x+4|\in \mathbb{N},\,\,\,\forall x \in \mathbb{Z}$