9.4. The sum of ten natural numbers is 1001. What is the greatest value that the GCD (greatest common divisor) of these numbers can take?
Answer: 91.

Solution. Example. Consider nine numbers equal to 91, and the number 182. Their sum is 1001.

Estimate. We will prove that the GCD cannot take a value greater than 91. Note that $1001=7 \times 11 \times 13$. Since each term in this sum is divisible by the GCD, the GCD is a divisor of the number 1001. On the other hand, the smallest term in the sum (and thus the GCD) is no greater than 101. It remains to note that 91 is the largest divisor of the number 1001 that satisfies this condition.

## Grading criteria:

+ fully justified solution

$\pm$ correct answer provided, estimate is proven, but no example is given

干 correct answer and an example of ten numbers are provided

- only the answer is provided
- the problem is not solved or is solved incorrectly