[tex]a+b= 72\\(a, b) = 12\Rightarrow \text{exist\u a dou\u a numere naturale } u \text{ \c si } v \text{, cu } (u, v) = 1\text{, astfel \^inc\^at } a=12u\text{ \c si } b=12v \\\Rightarrow 12u+12v=72\Rightarrow12(u+v) = 72\Rightarrow u+v=6[/tex]
Avem urmatoarele cazuri:
[tex]u = 1, \: v = 5: (1, 5) = 1\Rightarrow a=12,\: b=60;\\u = 2,\: v=4:(2,4)=2\neq1;\\u = 3,\: v = 3: (3, 3)\neq1\\u = 4,\: v = 2:(4, 2)\neq1\\u = 5\:, v = 1: (5, 1) = 1\Rightarrow a=60, \:b=12[/tex]
PRin urmare, numerele sunt 12 si 60.
