Question

15 Arătaţi că: a numărul A = 3^15 + 3^16 + 3^17 este divizibil cu 13;
Rezolvare:

b numărul B= 2^22 +2^24 +2^26 este divizibil cu 21.
Rezolvare:​

Answer 1

 

$\displaystyle\bf\\15.a)\\\\A=3^{15}+3^{16}+3^{17}=\\\\=3^{15}+3^{15+1}+3^{15+2}=\\\\=3^{15}+3^{15}\times3+3^{15}\times3^2=\\\\=3^{15}\Big(1+3+3^2\Big)=\\\\=\Big(1+3+9\Big)\times3^{15}=\\\\=13\times3^{15}~\vdots~13~deoarece~un~factor=13\\\\\implies~\boxed{\bf A~\vdots~13}$

.

$\displaystyle\bf\\15.b)\\\\B=2^{22}+2^{24}+2^{26}=\\\\=2^{22}+2^{22+2}+2^{22+4}=\\\\=2^{22}+2^{22}\times2^2+2^{22}\times2^4=\\\\=2^{22}\Big(1+2^2+2^4\Big)=\\\\=\Big(1+4+16\Big)\times2^{22}=\\\\=21\times2^{22}~\vdots~21~deoarece~un~factor=21\\\\\implies~\boxed{\bf B~\vdots~21}$