Răspuns:
[tex]\boldsymbol{ \red{11 \ \big| \ A}}[/tex]
Explicație:
8 = 2³, 125 = 5³, 4 = 2², 10 = 2 · 5
[tex]A = (2^3)^{n+1} \cdot 5^{3n} + 2^{3n+2} \cdot (5^3)^{n} - (2^2)^{n} \cdot 5^{2n} \cdot (2 \cdot 5)^{n} \\[/tex]
[tex]A = 2^{3(n+1)} \cdot 5^{3n} + 2^{3n+2} \cdot 5^{3n} - 2^{2n} \cdot 5^{2n} \cdot 2^{n} \cdot 5^{n}\\[/tex]
[tex]A = 2^{3n+3} \cdot 5^{3n} + 2^{3n+2} \cdot 5^{3n} - 2^{3n} \cdot 5^{3n}\\[/tex]
Vom da factor comun:
[tex]A = 2^{3n} \cdot 5^{3n} \cdot (2^3 + 2^{2} - 1)\\[/tex]
[tex]A = (2 \cdot 5)^{3n} \cdot (8 + 4 - 1)\\[/tex]
[tex]A = 11 \cdot 10^{3n} \implies \boldsymbol{ 11 \ \big| \ A}[/tex]
q.e.d.
