6. The sum of ten consecutive even natural numbers is 5,370. Determine all prime divisors of the largest of these numbers.
6. If $x$ is the first unknown number, the other even numbers are $x+2, x+4, x+6$, and so on...

The tenth in order is $x+18$.

By adding all these numbers, we get

$x+x+2+x+4+x+6+\ldots+x+18=5370$.

$1 \text{ POINT}$

It holds that

$2+4+6+\ldots+18=90$.

$1 \text{ POINT}$

Further, we have:

$10 x+90=5370$

$1 \text{ POINT}$

$10 x=5370-90$

$10 x=5280$

$x=5280: 10$

$x=528$

$1 \text{ POINT}$

The required numbers are: 528, 530, 532, ...,

and the largest (last) among them is 546.

From the factorization $546=2 \cdot 3 \cdot 7 \cdot 13 \quad 2 \text{ POINTS}$*

it follows:

The prime factors of the largest required number are: 2, 3, 7, and 13.

$1 \text{ POINT}$

TOTAL 10 POINTS

*Note: Students who do not notice that 91 is a composite number and leave $546=2 \cdot 3 \cdot 91$, should receive only 1 POINT instead of 2 POINTS here.