Question

Question #24690

Answer 1

Considering only positive solutions,

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

Explanation:

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

`15x+11y=457` because `2(15x+11y)=2 cdot457`

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

`ax+by=c` has integer solution if and only if `d = gcd(a,b)` 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.