a)
[tex]2 ^{n + 2} \times {3}^{n + 1 } + {2}^{n + 3} \times 3 ^{n} \\ 2 ^{n} \times {2}^{2} \times {3}^{n} \times 3^{1} + {2}^{n} \times {2}^{3} \times {3}^{n} \\ {2}^{n} \times {3}^{n} \times {2}^{2} \times 3 + {2}^{n} \times {3}^{n} \times 2 ^{3} [/tex]
observam ca putem aplica metoda factorului comun in cazul nostru 2^n×3^n
[tex] {2}^{n} \times {3}^{n} ( {2}^{2} \times 3 + {2}^{3} \times 1) \\ {2}^{n} \times {3}^{n} (4 \times 3 + 8) \\ {2}^{n} \times {3}^{n} \times 20[/tex]
20 divizibil cu 4=> ca tot numarul se divide cu 4
b)
[tex]{3}^{n + 1} \times {7}^{n + 1} - {3}^{n + 2} \times {7}^{n} \\ {3}^{n} \times {7}^{n} \times 3 \times 7 - {3}^{n} \times {7}^{n} \times {3}^{2} \\ {3}^{n} \times {7}^{n}(3 \times 7 - {3}^{2} ) \\ {3}^{n} \times {7}^{n} \times 4 \times 3[/tex]
deci numarul dat este multiplu de 4 deoarece se poate scrie ca X×4
cu draaag
