Question

(a,b)x[a,b]=864
(a,b)+[a,d]=84. a,b=?

Answer 1

ab=864
si ab= (a,b) * [a,b]
atunnci
(a,b) +ab/(a.b)=84
(a,b)+864/(a,b)=84

notam (a,b)=x
x+864/x=84  | *x
x²-84x+864=0

Δ=84²-4*864=3600
√Δ=60

x1,2=(84+-60)/2
x1=12
x2=72

(a,b) =12
[a,b]=864:12=72 convine


(a,b)=72
[a,b]=864:72=12, nu convine, pt ca [a,b]≥(a,b)


(a,b) =12⇒a=12n; b=12m unde (n,m)=1   (1)
[a,b]=72⇒[12n;12m]=72=12*6⇒[n,m]=6    (2)

din (1) si (2)⇒perechile (n;m)∈{ (1;6) ;(2;3);(3;2);(6;1)}
atunci  perechile (a;b)∈{(12;72);(24;36);(36.24);(72;12)}