Problem 5. Find all pairs of natural numbers $a$ and $b$ such that

$$
\operatorname{LCM}(a, b)=\text { GCD }(a, b)+19
$$

(and prove that there are no others).

GCD $(a, b)$ is the greatest common divisor, i.e., the largest natural number that divides both $a$ and $b$. $\operatorname{LCM}(a, b)$ is the least common multiple, i.e., the smallest natural number that is a multiple of both $a$ and $b$.

Answer: $(a, b)=(1,20),(20,1),(4,5),(5,4),(19,38),(38,19)$.
Solution. Let $d=\gcd(a, b)$. Note that both the LCM and the GCD are divisible by $d$, which means that 19 is also divisible by $d$. Since 19 is a prime number, we get that $d=1$ or $d=19$.

- If $d=1$, then the numbers $a$ and $b$ are coprime, and $\operatorname{lcm}(a, b)=a \cdot b=1+19=20$. This gives the options $(1,20),(20,1),(4,5),(5,4)$.
- If $d=19$, then $\operatorname{lcm}(a, b)=19+19=38$. This means that $a=19, b=38$ or $a=38, b=19$.


## Criteria

## 16 . The correct answer is provided.

2 p. Only one of the cases $d=1$ or $d=19$ is considered.

4 p. The correct answer and justification are provided.