Answer:
There are none.
Explanation:
gcd(431,29)=1
The reasons I came to this conclusion is because 29 is prime.
29 is not a factor of 431 so we are done.
So now we want to find (x,y) such that 115x+30y=1.
I'm going to use Euclidean's Algorithm.
115=30(3)+25
30=25(1)+5
25=5(5)
So we know we are done when we get the remainder is 0 and I like to look at the line before the remainder 0 line to see the greatest common divisor or 115 and 30 is 5.
So 115x+30y=5 has integer solutions (x,y) where d=5 is the smallest possible positive such that 115x+30y=d will have integer solutions (x,y).
So since 1 is smaller than 5 and we are trying to solve 115x+30y=1 for integer solutions (x,y), there there is none.
Furthermore, 115x+30y=1 can be written as 5(23x+6y)=1 and we know that 5 is not a factor of 1.