Find integers x and y such that 115x + 30y = gcd(431, 29)

There are none.

Step-by-step 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.

1

step-by-step explanation:

you are throwing one dart