Question

arătați că nr. N=3^n+2×5^n+1-7×15^n+2×15^n+1 este divizibil cu 17.

Answer 1

N=3^(n+2)×5^(n+1)-7×15^n+2×15^(n+1)=

=3^(n+2)×5^(n+1)-7×3^n×5^n+2×3^(n+1)×5^(n+1)=

=3^n×5^n×(3^2×5 -7+2×3^1×5^1)=

=3^n×5^n×(45 -7+30)=

=3^n×5^n×68=                                  68=2^2×17

=3^n×5^n×2^2×17 ⇒ divizibil cu 17