Let two numbers be $x$ and $y$.
According to the question sum of these two is equal to 31.
So, $x+y = 31$
Also, their product is equal to 238, i.e. $x \cdot y = 238$
To find $x$ and $y$ we need to solve the following equations:
$$x+y = 31$$

$$x \cdot y = 238$$

$$x+y = 31 => x = 31-y$$

Using this value of $x$ in the second equation, we get
$$(31-y) \cdot y = 238$$

$$31y-y^2 = 238$$

$$y^2-31y+238 = 0$$

Using the method to factor trinomials [8].

Here, a = 1, b = -31 and c = 238.
$a \cdot c = 238$ and $b = -31$.
Also, $238 = -14 \cdot -17$ and $-14-17 = -31$
This implies:
$$y^2-14y-17y+238 = 0$$

$$y(y-14)-17(y-14) = 0$$

$$(y-14)(y-17) = 0$$
Either $(y-14) = 0 => y = 14$ or (y-17) = 0 => y = 17$

If $y = 14$ then $x+14 = 31 => x = 17$

If $y = 17$ then $x+17 = 31 => x = 14$

So, our two numbers are 14 and 17.
The Common Multiples of 14 and 17 are 238, 476, 714, ...

Therefore the least common multiple (LCM) of 14 and 17 = 238