In our initial case we have 5x+3y=1 (or -1, we can reverse the signs of x and y to negate the equation, so this does not matter). Clearly this equation has solutions, one of which is (x,y)=(2,-3).

Now, in our equation ax+by=1 let the highest common factor of a and b be d with a=da¢ and b=db¢ (a¢ and b¢ both integers also). Our equation becomes d(a¢x+b¢y)=1. However, the only factors of 1 are ±1, so d=1. Therefore, for our equation to have any solutions, a and b cannot have any common factors (are coprime), so, for example, a=3, b=6 will not work because they both share factor 3. Interestingly, this means we can make (a¢x+b¢y) whatever value we choose, because a¢ and b¢ must be coprime (with their common factors extracted out as d), and we can choose x and y to make a¢x+b¢y=1 and then multiply them by anything to get any other integer.

We can extend this logic to the general form ax+by=c, so d(a¢x+b¢y)=c. Therefore, we will have solutions if and only if d is a factor of c (since a¢x+b¢y can be anything). I'll complete my solution with a few examples. 54x+36y=47 has no solutions, because 18 (the highest common factor of 36 and 54) does not divide 47. 30x+25y=10 has solutions because 5 does divide 10 (for example, x=2, y=-2). With actual numbers in a, b, c, the fact we established above becomes fairly obvious, because we can divide through by the highest common factor to get a¢ and b¢ (here 6 and 5), and because they are coprime, we can find x, y with 6x+5y=1 so multiplying x and y by (c/d) gives our solution.

This is an example of a ``linear Diophantine equation'', a concept in number theory.

Tom noticed that if a' and b' are coprime then we can find integers x and y such that a'x+b'y=1. This result is sometimes known as Bezout's Theorem; can you find a proof?