Bézout's identity
When you have two numbers, let's say 6 and 15, they both have different factors, or numbers that can divide into them without leaving a remainder. For 6, the factors are 1, 2, 3, and 6. For 15, the factors are 1, 3, 5, and 15.
Now if we try to find a number that is a common factor of both 6 and 15, meaning it can divide into both numbers without leaving a remainder, we find that 3 is that number. This is because 6 divided by 3 gives us 2 with no remainder, and 15 divided by 3 gives us 5 with no remainder.
Bézout’s identity is a fancy way of saying that when we have two numbers, there will always be a way to find a combination of them that equals their greatest common factor. In our example, the greatest common factor of 6 and 15 is 3. So Bézout’s identity tells us that there is a way to find two numbers that add up to 3 times some other number, and these two numbers are related to the original numbers.
In fact, Bézout’s identity says that there are always two numbers, let's call them A and B, for any pair of numbers where their greatest common factor is G. These two numbers can be found by using a bit of math, but the important thing to know is that they exist and are related to the original pair of numbers in a special way.
So, the next time you're looking for the greatest common factor of two numbers, remember that Bézout’s identity says there are always two other numbers that add up to it!
Now if we try to find a number that is a common factor of both 6 and 15, meaning it can divide into both numbers without leaving a remainder, we find that 3 is that number. This is because 6 divided by 3 gives us 2 with no remainder, and 15 divided by 3 gives us 5 with no remainder.
Bézout’s identity is a fancy way of saying that when we have two numbers, there will always be a way to find a combination of them that equals their greatest common factor. In our example, the greatest common factor of 6 and 15 is 3. So Bézout’s identity tells us that there is a way to find two numbers that add up to 3 times some other number, and these two numbers are related to the original numbers.
In fact, Bézout’s identity says that there are always two numbers, let's call them A and B, for any pair of numbers where their greatest common factor is G. These two numbers can be found by using a bit of math, but the important thing to know is that they exist and are related to the original pair of numbers in a special way.
So, the next time you're looking for the greatest common factor of two numbers, remember that Bézout’s identity says there are always two other numbers that add up to it!
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