[ equations in integers ]

Solve the equation $1990 x-173 y=11$ in integers.
As can be seen from problem 60514, it is sufficient to find a particular solution.

The first method. First, using the Euclidean algorithm, we find a solution to the equation $1990 x - 173 y = 1$ (see problem 60488), and then multiply the results by 11.

The second method. $1990 = 11 \cdot 173 + 87$, so the equation can be written as $173 z + 87 x = 11$, where $z = 11 x - y$. $173 = 2 \cdot 87 - 1$, so $87 t - z = 11$, where $t = 2 z + x$.

Setting $t_{0} = 0$, we get $z_{0} = -11, x_{0} = t_{0} - 2 z_{0} = 22, y_{0} = 11 x_{0} - z_{0} = 253$.

## Answer

$(22 + 173 k, 253 + 1990 k), k \in \mathbf{Z}$.

Solve the equation $2 x + 3 y + 3 z = 11$ in integers.

## Solution

Writing the equation as $3(x + y + z) = 11 + x$, we see that $11 + x$ is divisible by 3. Therefore, $x = 3 k + 1, x + y + z = k + 4, x + y = k + 4 - z - x = 3 - 2 k - z$.

## Answer

$(1 + 3 k, 3 - 2 k - l, l), k, l \in \mathbf{Z}$.