9.3. What is the smallest number of digits that need to be appended to the right of the number 2014 so that the resulting number is divisible by all natural numbers less than $10?$
Answer: 4 digits. Solution. If three digits are appended to 2014, the resulting number will be $\leq 2014$ 999. Dividing 2014999 by $2520=$ LCM $(1,2, \ldots, 9)$, we get a quotient of 799 and a remainder of 1519. Since $1519>1000$, there is no integer multiple of 2520 between the numbers 2014000 and 2014999. Therefore, it is impossible to append three digits to 2014. From this, it follows that it is also impossible to append one or two digits (otherwise, it would be possible to append zeros to make up three digits). Four digits will be sufficient, since $10000>2520$, and therefore there is an integer multiple of 2520 between 20140000 and 20149999 (the smallest of such numbers is $20142360=2520 \cdot 7993$).