Subiectul I
a, b numere naturale
a+b=42
(a,b)=7
daca doua numere au cmmdc =7 atunci putem scrie
a=7m
b=7n, iar m si n prime intre ele (m,n)=1
inlocuim in suma
7m+7n=42
7(m+n)=42
m+n=6 si (m,n)=1
m=1, n=5, a=7, b=35
m=5, n=1, a=35, b=7
b. x, y, z∈N
6(x+y)=8(x+z)=4(y+z)=k, k∈N
x+y+z=78
x+y=k/6
x+z=k/8
y+z=k/4
2(x+y+z)=4k/24+3k/24+6k/24
2·78=13k/24
13k=2·78·24
k=2·6·24
k=288
x+y=288/6=48
x+z=288/8=36
y+z=288/4=72
x+y=48
x+y+z=78
z=30
y+30=72, y=42
x+42=48, x=6
verificare 6+42+30=78
numerele sunt 6, 42, 30.
