Bézout's theorem

Imagine you have two numbers and you want to find their biggest common factor, which is the biggest number that can be divided evenly by both of them. For example, if you have 12 and 18, their biggest common factor is 6.

Now imagine you have a big sheet of paper and you draw a line for each of your two numbers, starting at 0 and going up to that number. So for 12 you draw a line that goes from 0 to 12, and for 18 you draw a line from 0 to 18.

Next, you see where the two lines intersect. That point is going to be a multiple of the biggest common factor, because it's where the two numbers have a common divisor.

Bézout's Theorem says that not only is that point a multiple of the biggest common factor, but you can also use math to find two other numbers (let's call them "X" and "Y") that, when you multiply them by the two original numbers and add them together, you get that point where the lines intersect.

So if we go back to our example of 12 and 18, we know that their biggest common factor is 6, and we can see that the point where their lines intersect is 6.

So what two numbers can we multiply the original numbers by, and then add together, to get 6? It turns out, we can multiply 12 by -1 and multiply 18 by 1, and when we add those together, we get 6.

So Bézout's Theorem is a way to find two numbers that help us solve certain math problems involving common divisors, by looking at two lines on a piece of paper and finding where they intersect.