1. Let $\mathrm{d}=(a, b)$ and $m=[a, b]$ then:

$a=d x, b=d y, (x, y)=1$ and $5 d x+3 d y=150$ and $m=15 d$.

But $d m=a b$, so $m=d x y$ and since $m=15 d$, it follows that $x y=15$.

We have $(x, y) \in\{(1,15) ;(15,1) ;(3,5) ;(5,3)\}$

For $(x, y) \in\{(15,1) ;(5,3)\} \Rightarrow d \notin \mathbb{N}$.

For $(x, y)=(1,15)$, it follows that $d=3, a=3, b=45$

For $(x, y)=(3,5)$, it follows that $d=5, a=15, b=25$
1. (7p) Determine the natural numbers $a$ and $b$ whose least common multiple is 15 times greater than their greatest common divisor and $5a + 3b = 150$.

GMB2012