Question

a) daca 2a+3b=65 si (a,b)=5, atunci determinati numerele naturale a si b
b)daca a*b=294 si (a,b)=7, atunci determinati numerele naturale a si b

Answer 1

a)
(a,b)=5 => a= 5x si b = 5y cu (x,y)=1
2*5x + 3*5y = 65 | :5 => 2x + 3y = 13

Cazul 1: y=0 => 2x = 13 NU
Cazul 2 : y=1 => 2x = 13-3 =10 |:2 => x=5 => a=5*5=25 si b=5y=>b=5*1=5
Cazul 3 : y=2 => 2x= 13-6=7 NU
Cazul 4 : y=3 => 2x = 13-9 = 4 |:2 => x=2 => a=5*2=10 si b=5*3=15
Cazul 5 : y=4 => 2x=13-12=1 NU.
(a;b)∈{(25;5) ; (10;15)}
b)
(a,b) = 7 => a= 7x si b=7y cu (x,y) =1
a*b=294 => 7x*7y =294 => x*y * 49 =294 => x*y = 6
Cazul 1 : x=1 , y =6 => a= 7*1=7 si b=7*6=42
Cazul 2 : x=2 , y=3 => a= 7*2 = 14 si b= 7*3=21
Cazul 3 : x=3 , y=2 => a=7*3=21 si b=7*2=14
Cazul 4 : x=6 , y=1 => a=7*6=42 si b=7*1=7
(a,b) ∈ { (7,42) , (14,21) , (21,14) , (42,7) }
Sper ca ai inteles . Succes in continuare !