2. The difference between two natural numbers is 15, and their least common multiple is 42. What are these numbers? Explain your answer!
Solution. If we denote one of these numbers by $a$, then the other number we are looking for is $a+15$. The LCM of these two numbers is 42, so both numbers are divisors of 42. Therefore, the number $a$ is one of the numbers $1, 2, 3, 6, 7$, 14, 21, or 42. However, the number $a+15$ is also one of these numbers. Since $a+15$ is greater than 15 and is a divisor of 42, we have:
i) $a+15=21$, from which $a=6$;

ii) $a+15=42$, from which $a=27$, which contradicts the condition that the number $a$ is a divisor of 42.

Therefore, the numbers we are looking for are 6 and 21.