27 The difference between the highest common factor and the lowest common multiple of $x$ and 18 is 120 . Find the value of $x$.
27. Answer: 42
Let the HCF of $x \& 18$ be $k$ and let $x=k a, 18=k b$ where $\operatorname{gcd}(a, b)=1$. We have LCM of $x \& 18=k a b$. From information given, $k a b-k=120$. Clearly gcd $(a b-$ $1, b)=1$, so $\operatorname{gcd}(k a b-k, k b)=k \Rightarrow \operatorname{gcd}(120,18)=k \Rightarrow k=6$. Hence $b=3$ and . $x=42$.