4. Determine all pairs of natural numbers $(a, b)$ such that $(a+6)(b+5)=3ab$.

Find all pairs of natural numbers $(a, b)$ that satisfy the equation $(a+6)(b+5)=3ab$.
First solution.

$a b+5 a+6 b+30-3 a b=0$

$1 \text{POINT}$

$-2 a b+5 a=-6 b-30 / \cdot(-1)$

$2 a b-5 a=6 b+30$

$a(2 b-5)=6 b+30$

$1 \text{POINT}$

$a=\frac{6 b+30}{2 b-5}$

$a=\frac{3(2 b-5)+15+30}{2 b-5}$

$a=3+\frac{45}{2 b-5}$

$1 \text{POINT}$

$a \in \mathbb{N} \Leftrightarrow \frac{45}{2 b-5} \in \mathbb{N} \Leftrightarrow 2 b-5 \in\{1,3,5,9,15,45\}$

2 POINTS

$\Leftrightarrow 2 b \in\{6,8,10,14,20,50\}$

$\Leftrightarrow b \in\{3,4,5,7,10,25\} \quad 1$ POINT

$\Leftrightarrow \frac{45}{2 b-5} \in\{45,15,9,5,3,1\} \quad 1$ POINT

$\Leftrightarrow a=3+\frac{45}{2 b-5} \in\{48,18,12,8,6,4\} \quad 1$ POINT

The sought pairs of numbers are $(48,3),(18,4),(12,5),(8,7),(6,10),(4,25)$.

2 POINTS

TOTAL 10 POINTS

Note: If all ordered pairs are provided in the answer, the student receives 2 POINTS; if three, four, or five ordered pairs are provided, the student receives 1 POINT; if fewer than three ordered pairs are provided, the student receives 0 POINTS.