The difference between the HCF and LCM of $x$ and $18$ is $120$. Find $x$.

No casework needed.

$\gcd(x,18) + 120 = \text{lcm} (x,18) \implies \gcd(x,18) \equiv -120 \equiv 6 \pmod{18}$

Therefore $\gcd(x,18)=6, x = \gcd(x,18)\cdot \text{lcm}(x,18)/18 = 6 \cdot 126/18=42$.

Since $y\le 18,y=6,12,18$. Reject $12$ since it is not a factor of $18$.

We get corresponding $z=120+y=126,138$. Reject $138$ since it is not a multiple of $18$.

We are left with $y=6,z=126,x=yz/18=42$.