7,8,9 |
| $[\underline{\text { Modular Arithmetic (other) }}]$ |  |  |

Find all prime numbers that are equal to the sum of two prime numbers and the difference of two prime numbers.
The given prime number $p$ is odd, so the sum and difference involve numbers of different parity. Thus, $p=q+2=r-2$. From this, it is clear that the numbers give different remainders when divided by 3, meaning one of them is divisible by 3, and since it is prime, it equals 3.

## Answer

5. 

![](https://cdn.mathpix.com/cropped/2024_05_06_5008c33ea747074d348fg-21.jpg?height=54&width=1032&top_left_y=57&top_left_x=17)

Difficulty: 3 [ Residue arithmetic (other) $]$

In a right-angled triangle, the lengths of the sides are natural coprime numbers.

Prove that the length of the hypotenuse is an odd number, and the lengths of the legs have different parity.

## Hint

If $a$ and $b$ are odd numbers, then the equation $a^{2}+b^{2}=c^{2}$ is impossible, since $c^{2}$ cannot give a remainder of 2 when divided by 4.