Question

fie a,b,c nr prime care verifica egalitatea: a(a+b)+b(b+c)+c(c+a)=69 daca a

Answer 1

a²+ab+b²+bc+c²+ac=69 l *2
2a²+2ab+2b²+2bc+2c²+2ac=138
(a²+2ab+b²) +(a²+2ac+c²) +(b²+2bc+c²) =138
(a+b)²+(a+c)²+(b+c)²=138
11²=121
12²=144
121<138<144 ; 138 nu e pp
Notam x=a+b
Y= a+c
Z=b+c
x²+y²+z²=138
Pp <138 sunt : 1 4 9 16 25 36 49 64 81 100 121
Ceea ce inseamna ca o paranteza poate avea valoarea : 1;2;3;4;5;6;7;8;9;10;11
Nr prime intre 1 si 11 sunt :2;3;5;7;11
2+11=13
13²=169=> 11 nu poate fii luat in considerare=>
a, b, c pot avea valori intre : 2;3;5;7
1.a=2;b=3;c=5
(2+3)²+(2+5)²+(3+5)²=5²+7²+8²=25+49+64=138=>Nr cautate sunt : 5;7;8