Question

determinati numerele naturale nenule a si b pentru care
a) 2 x [a,b] -5 x (a,b) = 306
b) [a,b] = 28 x (a,b)

Answer 1

Fie (a,b) =d => a=dx ; b=dy, unde (x,y)=1

[a,b]=d*x*y

a) 2 * [a,b] -5 * (a,b) = 306

2*d*x*y - 5*d = 306

d(2*x*y-5)=306

2*x*y=par, 5=impar => 2*x*y-5=impar , 306=par =>d=par

Fie d=2k

2k(2*x*y-5)=306 => k(2*x*y-5)=152

152=8*19

Cum am spus mai sus, 2*x*y-5=impar. Deci 2*x*y-5=19 => 2*x*y = 24 => x*y = 12

12=3*4 , (3,4)=1 , x=3, y=4

Si k=8=> d=16

Asadar a=3*16 => a=48 si b=4*16 => b=64

b) [a,b] = 28 * (a,b)

d*x*y = 28*d => x*y =28

28= 7*4 ; (7,4)=1 => x=7 si y=4  

Deci a=7 si b=4