[tex]\it 7\cdot6^{2n}\cdot5^{n+1}-6\cdot4^n\cdot3^{2n}\cdot5^n=7\cdot2^{2n}\cdot3^{2n}\cdot5^n\cdot5-6\cdot4^n\cdot9^n\cdot5^n=\\ \\ =35\cdot4^n\cdot9^n\cdot5^n-6\cdot4^n\cdot9^n\cdot5^n=35\cdot180^n-6\cdot180^n=29\cdot180^n[/tex]
[tex]\it 19\cdot3^n\cdot2^n\cdot5^{n-1}+2\cdot10^n\cdot3^n=19\cdot3^n\cdot2^n\cdot5^n\cdot5^{-1}+2\cdot(10\cdot3)^n=\\ \\ =19\cdot\dfrac{1}{5}\cdot30^n+2\cdot30^n=\ \Big(\dfrac{19}{5}+2\Big)\cdot30^n=\dfrac{29}{5}\cdot30^n[/tex]
[tex]\it N=\dfrac{29\cdot180^n}{\dfrac{29}{5}\cdot30^n}=\dfrac{29}{\dfrac{29}{5}}\cdot\Big(\dfrac{180}{30}\Big)^n=29\cdot\dfrac{5}{29}\cdot6^n=5\cdot6^n\in\mathbb{N}[/tex]
