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1. a) Stiind ca a=(1+2+3+4+....+2015)-(11+12+13+...+2015 ) , aratati ca nr (a+45)x(a-45)se imparte exact la 1000 .Rezolvare completa 

Răspuns :

Dupa cum putem observa se pot anula(scadea) termenii 11,12,...,2015(ca in prima paranteza ii adunam si in a doua ii scadem, deci ramanem cu 0 => a=1+2+...+10=55. => (a+45)*(a+45)=100*100=10000 care se imparte evident la 1000(da rezultatul 10.)
a=1+2+3+4+...+2015-11-12-13-...=2015=1+2+3+4+...+10=55
(a+45)x(a-45)=(55+45)x(55-45)=100x10=1000  se imparte exacy la 1000