Question

1.Aratati ca numatul
a=30+$2^{2015}$ se divide cu 62.

Answer 1

[tex]a=62-32+2^{2015}\\ a=62-2^5+2^{2015}=62+2^5(2^{2010}-1)=62+2\cdot 16\cdot (2^{2010}-1)\\ Aratam\ ca (2^{2010}-1)\vdots31.\\ 2^{2010}-1=(2^{5})^{402}-1=32^{402}-1^{402}=(32-1)(32^{401}+...+1)=31k\\ a=62+62t=62(1+t)=\ \textgreater \ a\vdots62.[/tex]