# Question #24690

Nov 22, 2016

Considering only positive solutions,

$x = 7 , y = 32$
$x = 18 , y = 17$ and
$x = 29 , y = 2$

#### Explanation:

$30 x + 22 y = 914$ has the same solutions of

$15 x + 11 y = 457$ because $2 \left(15 x + 11 y\right) = 2 \cdot 457$

Now, by Bézout's identity,
https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity

$a x + b y = c$ has integer solution if and only if $d = \gcd \left(a , b\right)$ is a divisor of $c$

In our case, $d = 1$ and $c = 457$ is prime, but $1$ is a universal divisor so the equation must have solutions.

$x = 7 , y = 32$
$x = 18 , y = 17$ and
$x = 29 , y = 2$

Considering only positive solutions.