2. Determine all pairs of natural numbers $(a, b)$ such that $a+b=378$ and $D(a, b)=63$

(8) ( $D(a, b)$ is the greatest common divisor).
Solution. From $a=63 x, b=63 y$ we get $63 x+63 y=378$ and $x+y=6, x, y \in \mathbb{N}$, $D(x, y)=1$.

Then $x \in\{1,5\}$ and $y=6-x$ so $(a, b) \in\{(63,315),(315,63)\}$

(for a complete solution 8 points)

(if it writes $x \in\{2,4\}$ and $(a, b) \in\{126,252),(252,126)\}$ or

$x \in\{3,3\}$ and $(a, b) \in\{189,189)\}, 4$ points $)$