MaBloWriMo 24: Bezout’s identity

A few days ago we made use of Bézout’s Identity, which states that if a and b have a greatest common divisor d, then there exist integers x and y such that ax + by = d. For completeness, let’s prove it.

Consider the set of all linear combinations of a and b, that is,

\{ax + by \mid x, y \in \mathbb{Z} \},

and suppose d = as + bt is the smallest positive integer in this set. For example, if a = 10 and b = 6, then you can check that, for example, 16 = a + b, and 8 = 2a - 2b, and 4 = a - b are all in this set, as is, for example, -4 = -a + b, but the smallest positive integer you can get is 2 = 2a - 3b. We will prove that in fact, d is the greatest common divisor of a and b.

Consider dividing a by d. This will result in some remainder r such that 0 \leq r < d. I claim the remainder is also of the form ax + by for some integers x and y: note that a = a1 + b0 is of this form, and d is of this form by definition, and we get the remainder by subtracting some number of copies of d from a. Subtracting two numbers of the form ax + by works by subtracting coefficients, yielding another number of the same form again. But d is supposed to be the smallest positive number of this form, and r is less than d—which means r has to be zero, that is, d evenly divides a. The same argument shows that d evenly divides b as well. So d is a common divisor of a and b.

To see that d is the greatest common divisor, suppose c also divides a and b. Then since d = ax + by, we can see that c must divide d as well—so it must be less than or equal to d.

Voila! This proof doesn’t show us how to actually compute some appropriate x and y given a and b—that can be done using the extended Euclidean algorithm; perhaps I’ll write about that some other time. But this proof will do for today.

For the remainder of the month, as suggested by commented janhrcek, we’ll prove the thing I hinted at in an earlier post: namely, that the order of any group element is always a divisor of the order of the group. This is a really cool proof that hints at some much deeper group theory.

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About Brent

Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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2 Responses to MaBloWriMo 24: Bezout’s identity

  1. Logan says:

    When showing that d is a common divisor, you state “…that is, d evenly divides a.” The next sentence states “…shows that d evenly divided a as well.” Shouldn’t the second be b?

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