## PROBLEM 1

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

Mathematical Gazette
## Solution:

Let $(a, b)=d$, then $a=d \cdot x, b=d \cdot y$, where $x, y \in N$ and $(x, y)=1$........................... 1 point

From $a \cdot b=(a, b) \cdot[a, b]$ and $[a, b]=15 \cdot d$, we get $d \cdot x \cdot d \cdot y=15 \cdot d^2 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots 2$ points

$x \cdot y=15 \Rightarrow (x, y) \in\{(1 ; 15) ;(15 ; 1) ;(3 ; 5) ;(5 ; 3)\}$............................................. 2 points

By substituting into the relation $5a+3b=150$ we get $d \in\{3 ; 5\}$......................................... 1 point

Finalization: $(a, b) \in\{(3 ; 45) ;(15 ; 45)\}$...................................................... 1 point