Question

Fie z,y ∈ N* astfel incat$\frac{1}{x}+ \frac{1}{y} = \frac{1}{2009}.$ Aratam ca √($\frac{x}{41}-49) ( \frac{y}{41}-49)$ ∈ N

Answer 1

Aducem la acelasi numitor in prima expresie si =>(x+y)/x*y=1/2009=>. x*y=2009*(x+y). facem inmultirea in radical si=>radical din(x*y/41 la patrat-49*x/41-49*y/41+49 la patrat)= radicak din(2009*(x+y)/41 la patrat-49/41(x+y)+49 la patrat) simolificam 2009 cu 41 la patart prin 41 si obtinem radical dun(49/41(x+y)-49/41(x+y)+49 la patrat). Se reduc primii termeni=> radu al din 49 la patrat= 49.