6. (20 points) Is the number 390629 prime? Provide a justified answer (without using a calculator).
Solution. Since $25^{4}=390625$, then

$390629=25^{4}+4=25^{4}+4+4 \cdot 25^{2}-4 \cdot 25^{2}=\left(25^{2}+2\right)^{2}-50^{2}=577 \cdot 677$.

Second method.

$390629 \cdot 256=10001024=10^{8}+2^{10}=2^{8}\left(5^{8}+4\right) ; 390629=\left(5^{4}+2\right)^{2}-50^{2}=577 \cdot 677$.

Answer: No.

| Content of the criterion | Points |
| :--- | :---: |
| The correct answer is obtained with justification. | 20 |
| The correct answer is obtained, but the solution is insufficiently justified. | 14 |
| The beginning of a possible solution is provided, some intermediate results are obtained, further solution is incorrect or missing. | 6 |
| The solution does not meet any of the criteria listed above. | 0 |
| Maximum score | 20 |