7.2. What is the smallest number of digits that can be appended to the right of the number 2013 so that the resulting number is divisible by all natural numbers less than 10?
Answer: three digits. Solution. The least common multiple of the numbers $(1,2, \ldots, 9)$ is 2520. If two digits are appended to 2013, the resulting number will not exceed 201399. Dividing 201399 by 2520 with a remainder, we get a remainder of 2319. Since 2319 > 99, there is no integer multiple of 2520 between the numbers 201300 and 201399. Therefore, it is impossible to append two digits to 2013. Clearly, it is also impossible to append one digit (otherwise, it would be possible to add another zero at the end). Three digits will suffice; indeed, when dividing 2013999 by 2520, the remainder is 519, and since $519<999$, we find the required number $2013999-519=$ 2013480, which is a multiple of 2520.