Question

Daca numerele intregi a,b,c verifica relatia ab + 5*c*(a-b) - 25*c^2 = 7
aratati ca |a+b|=8

Answer 1

ab+5c(a-b)-25c²=ab+5ac-5cb-25c²=a(b+5c)-5c(b+5c)=(b+5c)(a-5c)
7=1·7=-1·(-7)
Suma dintre cei 2 factori intregi al caror produs este 7 este fie 1+7=8, fie -1+(-7)=-8
Pe de alta parte, suma dintre aceiaisi doi factori este b+5c+a-5c=a+b, de unde rezulta ca a+b=8 sau a+b=-8. Concluzionam ca |a+b|=8.

Answer 2

$ab+5c(a-b)-25c^2=7 \\ \\ ab+5ac-5bc-25c^2=7 \\ \\ a(b+5c)-5c(b+5c)=7 \\ \\ (a-5c)(b+5c)=7 \\ \\ a,b \in Z \Rightarrow (a-5c) \in Z ~si~(b+5c) \in Z. \\ \\ 7=1 \cdot 7=(-1) \cdot (-7)$

$Deci~avem~4~cazuri: \\ \\ Cazul~I:~a-5c=1~si~b+5c=7. \\ \\ a+b=(a-5c)+(b+5c)=1+7=8 \Rightarrow |a+b|=8. \\ \\ Cazul~II:~a-5c=7~si~b+5c=1. \\ \\ a+b=(a-5c)+(b+5c)=7+1=8 \Rightarrow |a+b|=8. \\ \\ Cazul~III:~a-5c=-1~si~b+5c=-7. \\ \\ a+b=(a-5c)+(b+5c)=-1-7=-8 \Rightarrow |a+b|=8.$

$Cazul~IV:~a+b=(a-5c)+(b+5c)=-7-1=-8 \Rightarrow |a+b|=8. \\ \\ In~concluzie,~|a+b|=8.$