Question

determinati perechile de nr naturale(a,b),stiind ca:a)(a,b)=12
a ori b=864
b)[a,b]=315 a ori b =2835
d)a+b=120 (a,b)=15
e)3a+5b=105 (a,b)=5

Answer 1

a=12x, b=12y cu (x,y)=1
a*b=12x*12y=144*x*y=864
⇒x*y=864/144=6
x=2 si y=3 ⇒ a=12*2=24, b=1283=36

b. (a,b)[a,b]=a*b
(a,b)*315=2835
(a,b)=9
a=9x, b=9y cu (x,y)=1
9x*9y=2835
81*xy=2835
xy=2835:81=35
x=5, y=7
a=9*5=45, b=9*7=63

d. a=15x, b=15y cu  x si y prime intre ele adica (x,y)=1
15x+15y=120
15(x+y)=120
x+y=120:15
x+y=8
Sunt 2 solutii si anume
x=1, y=7 , a=15, b=15*7=105
x=3, y=5, a=3*15=45, b=5*15=75

a=5x, b=5y cu (x,y)=1
3a+5b=105
3*5x+5*5y=105
impart prin 5
3x+5y=21
x=2, 3*2+5y=21, 5y=21-6=15, y=3
a=2*5=10, b=3*5=15