Let $a$ and $b$ be natural numbers such that the greatest common divisor of $a$ and $b$ is 3. We are to find the natural numbers $a$ and $b$ such that
$$3 a - 2 b = \ell + 3$$
is satisfied, where $\ell$ is the least common multiple of $a$ and $b$.
When we set $a = 3 p$ and $b = 3 q$, the natural numbers $p$ and $q$ are mutually prime (co-prime), and hence $\ell = \mathbf { N } p q$. Thus using $p$ and $q$, the equality (1) can be transformed to
$$p q - \mathbf { O } p + \mathbf { P } q + \mathbf { Q } = 0 .$$
This can be further transformed to
$$( p + \mathbf { R } ) ( q - \mathbf { S } ) = - \mathbf { S } \mathbf { T } .$$
Among the pairs of integers $p$ and $q$ which satisfy this equation, the pair such that both $a$ and $b$ are natural numbers is
$$p = \mathbf { U } , \quad q = \mathbf { V } ,$$
which gives
$$a = \mathbf { W X } , \quad b = \mathbf { Y } .$$