Question

n=3^1+3^2+...+3^999.Aratati ca n e divizibil cu13
^=la putere

Answer 1

notez Mx = multiplu de x
daca N =3^1 +...+3^999 =>
=> 3*N =3*(3^1 +...+3^999 )=>
=>3*N=3^2+3^3+...+3^999+3^1000=>
=> 3*N-N =3^2+3^3+...+3^999+3^1000-(3^1 +...+3^999 )
=>2*N=3^1000-3^1=> N=(3^1000-3)/2  
cum (2,13) =1 (adica sunt prime intre ele) trebuie sa arati ca (3^1000-3) este divizibil cu 13
cum 3^1000-3 =3* 3^999-3 =3 * (3^3)^333 -3 =
=3* 27^333-3 =3*(2*13+1)^333-3=
=3*(M13+1)^333 -3 =3*M13 +3-3=3*M13 si evident este divizibil cu 13