## Subject (3).

a) Determine the set $M=\{(x, y) \in \mathbb{Z} \times \mathbb{Z} \mid 5 x+7 y=1\}$.

b) Determine the largest natural number $n$ that cannot be written in the form $n=5 x+7 y$, where $x, y \in \mathbb{N}$. Justify your answer.

Maria Pop, Cluj
Solution. It's almost not worth mentioning... How can such a problem be signed? I assume the authors of the two problems presented here are related; perhaps more care should be taken to avoid such things.

a) It is a classic result that if $p, q \in \mathbb{Z}^{*} \times \mathbb{Z}^{*}$ with greatest common divisor (gcd) $(p, q)=1$, then there exists $(x, y) \in \mathbb{Z} \times \mathbb{Z}$ such that $p x + q y = 1$, and specifically, if $\left(x_{0}, y_{0}\right)$ is a solution, then there exists $m \in \mathbb{Z}$ for which $x = x_{0} - m q$ and $y = y_{0} + m p$. Since $\left(x_{0}, y_{0}\right) = (3, -2)$ is a solution for $(p, q) = (5, 7)$, it means that

$$
M = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} \mid 5 x + 7 y = 1\} = \{(3 - 7 m, -2 + 5 m) \mid m \in \mathbb{Z}\}
$$

b) It is the same classic result, known as SYLVESTER, Frobenius, or the "Chicken McNugget" theorem, which states that under the above conditions, the largest natural number $n$ that cannot be written in the form $n = p x + q y$, with $x, y \in \mathbb{N}$, is $n = p q - p - q$ (and thus all numbers $n \geq (p-1)(q-1)$ can be represented in this way). Therefore, in our case, $n = 23$.

Remark. The results are as expected under these conditions

| Points | $7-6$ | $5-4$ | $3-2$ | $1-0$ |
| :---: | :---: | :---: | :---: | :---: |
| \# Scores |  | 3 | 4 | 100 |

Such theoretical results, or are known to competitors (which does not seem to be the case here), or have a minimal chance of being "discovered" in the heat of battle, during a competition.