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Se dau fractiile: Determinati a,b,c in Z stiind ca A,B,C sunt simultan numere intregi

O sa restructurez datele problemei

\( A=\frac{2004a+2005c}{2005b+2004c} \)

\( B=\frac{2004b+2005c}{2005c+2004a} \)

\( C=\frac{2004c+2005b}{2005c+2004b} \)

Cred ca iese destul de usor cu divizibiliate.
Cum \( A \in Z => (2005b+2004c) | (2004a+2005c) \)

\( B \in Z => (2005c+2004a) | (2004b+2005c) \)

\( C \in Z => (2005c+2004b) | (2004b+2005c) \)

Folosim proprietatea divizibilitatii \( a|a \)

\( => (2005b+2004c) | (2005b+2004c) \)
\( => (2005c+2004a) | (2005c+2004a) \)
\( => (2005c+2004b) | (2005c+2004b) \)

Scadem relatiile s.a.m.d. si ar trebui sa rezulte niste solutii simetrice de tipul
\( a=1 \)
\( b=1 \)
\( c=1 \)

MariusG wrote:Se dau fractiile:

Code: Select all

A=(2004a+2005c)/(2005b+2004c);
B=(2004b+2005c)/(2005c+2004a);
C=(2004c+2005b)/(2005c+2004b);

Determinati a,b,c in Z stiind ca A,B,C sunt simultan numere intregi

MariusG, invata si tu LaTeX - ul. Era chiar mai simplu de scris. Nu mai vorbesc de "frumos".

MariusG wrote:Se dau fractiile \( A=\frac {2004a+2005c}{2005b+2004c}\ ,\ B=\frac {2004b+2005c}{2005c+2004a}\ ,\ C=\frac {2004c+2005b}{2005c+2004b}\ . \)

Determinati numerele intregi \( a\ ,\ b\ ,\ c \) stiind ca \( A ,\ B\ ,\ C \) sunt simultan numere intregi.

Al3xx wrote:O sa restructurez datele problemei

\( A=\frac{2004a+2005c}{2005b+2004c} \)

\( B=\frac{2004b+2005c}{2005c+2004a} \)

\( C=\frac{2004c+2005b}{2005c+2004b} \)

Cred ca iese destul de usor cu divizibiliate.
Cum \( A \in Z => (2005b+2004c) | (2004a+2005c) \)

\( B \in Z => (2005c+2004a) | (2004b+2005c) \)

\( C \in Z => (2005c+2004b) | (2004b+2005c) \)

Folosim proprietatea divizibilitatii \( a|a \)

\( => (2005b+2004c) | (2005b+2004c) \)
\( => (2005c+2004a) | (2005c+2004a) \)
\( => (2005c+2004b) | (2005c+2004b) \)

Scadem relatiile s.a.m.d. si ar trebui sa rezulte niste solutii simetrice de tipul
\( a=1 \)
\( b=1 \)
\( c=1 \)

Mersi dar am incercat si eu dar nu a mers.
E undeva o lista cu comenzi LaTeX?