7.1. Write the number 2013 several times in a row so that the resulting number is divisible by 9. Explain your answer.
Answer: for example, 201320132013.

Solution. We will provide several ways to justify this.

First method. The sum of the digits of the written number is $(2+0+1+3) \cdot 3=18$, so it is divisible by 9 (by the divisibility rule for 9).

Second method. The sum of the digits of the number 2013 is divisible by 3, so if we write 2013 three times in a row, the sum of the digits of the resulting number will be divisible by $3 \cdot 3=9$, meaning the number itself will be divisible by 9 (by the divisibility rule for 9).

Third method. Dividing the written number by 9, we get: $201320132013: 9=22368903557$.

Fourth method. Dividing the written number by 3, we get: $201320132013: 3=67106710671$. The resulting number is divisible by 3, as the sum of its digits is 42. Therefore, the original number is divisible by 9.

There are other examples: we can write the number 2013 six times in a row, nine times in a row, and so on (any number of times that is a multiple of three). The justifications are similar to those provided above.

# Grading criteria:

+ one or more correct answers are provided and explained correctly

$\pm$ along with the correct answer, an incorrect one is also provided, but the correct answer is explained

干 the correct answer is provided without explanation or with an incorrect explanation

- only an incorrect answer is provided