Subject 2. Find the natural numbers $a$ and $b$ knowing that $[a, b]$ is 15 times larger than $(a, b)$ and $5 a+3 b=150$. We have denoted by $[a, b]$ the least common multiple and by $(a, b)$ the greatest common divisor of the numbers $a$ and $b$.
## Subject 2.

| Let $(a, b)=d \Rightarrow a=d \cdot x, b=d \cdot y,(x, y)=1 \quad x, y \in \mathbb{N}$ | $\mathbf{1 p}$ |
| :--- | :--- | :--- |
| $(a, b) \cdot[a, b]=a \cdot b \Rightarrow d \cdot 15 d=d \cdot x \cdot d \cdot y$ | $\mathbf{2 p}$ |
| We obtain $x \cdot y=15 \Rightarrow(x ; y) \in\{(1 ; 15) ;(15 ; 1) ;(3 ; 5) ;(5 ; 3)\}$ | $\mathbf{2 p}$ |
| Using the relation $5 a+3 b=150$ we get $d \in\{3 ; 5\}$ | $\mathbf{1 p}$ |
| Finalization: $(a ; b) \in\{(3 ; 45) ;(15 ; 45)\}$ | $\mathbf{1 p}$ |