3. A natural number $n>5$ is called new if there exists a number that is not divisible by $n$, but is divisible by all natural numbers less than $n$. What is the maximum number of consecutive numbers that can be new?
Solution. Answer: 3.

Example: the number 7 is new (60 is divisible by the numbers from 1 to 6, but not by 7);

the number 8 is new (420 is divisible by the numbers from 1 to 7, but not by 8);

the number 9 is new (840 is divisible by the numbers from 1 to 8, but not by 9).

Evaluation: every fourth number has the form $n=4k+2=2(2k+1)$; if some number is divisible by 2 and $2k+1$, then it is also divisible by $2(2k+1)$, therefore such an $n$ cannot be new.