14310

14310 (Traian Covaciu)

Fie Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n, n + 2, n + 6 } trei numere naturale și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S } suma lor.

a) Dați exemplu de cel puțin trei valori pentru Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb{N}} astfel incat numerele Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n, n + 2, n + 6 } sa fie simultan numere prime.

b) Daca $n,n+2,n+6$ sunt simultan numere prime, aratati ca exista $k\in \mathbb {N}$ astfel incat $S=9k+5$.

c) Daca $n,n+2,n+6$ sunt numere prime, determinati restul impartirii numarului $S$ la $18$.

Solutie:

a) Pentru $n=5$ numerele sunt $5,7,11$. Pentru $n=11$ avem $11,13,17$, iar pentru $n=17$ obtinem $17,19,23$.

b) Se stie ca numerele prime au forma $6p+1$ sau $6p+5$. Daca $n=6p+1$, atunci $n+2=6p+3$ care nu este numar prim. Asadar, $n=6p+5$. In aceasta situatie avem $S=6p+5+6p+7+6p+11=18p+23=9(2p+2)+5$. In concluzie, exista Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = 2p + 2 } astfel incat Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = 9k + 5 } .

c) Din punctul b) avem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = 18(p + 1) + 5 } , deci restul impartirii lui Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S } la Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 18 } este Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5 } .