Math 261
Spring 2014
GCD-Linear Combination Connection
Theorem Let a and b be integers. If (a, b) = 1, then there exist integers x and y such that
ax + by = 1.
We can easily see how this works with specific numbers. For example, (5, 3) = 1. By guess
and check, we can set x = 2 and y = −3 to see that 5x + 3y = 1.
A better way to do this is to use the The Euclidean Algorithm. We can use the pieces
from that theorem:
5 = 1 · 3 + 2 and 3 = 1 · 2 + 1
Starting with the right equation, we get 1 = 3 − 1 · 2. This is an equation with 1 equal to a
linear combination of 2 and 3. We need an equation that is 1 equal to a linear combination
of 3 and 5.
Use the left equation above to write 2 = 5 − 1 · 3. Then
1 = 3 − 1 · 2 = 1 = 3 − 1 · (5 − 1 · 3) = 2 · 3 − 1 · 5 = −1 · 5 + 2 · 3
and x = −1, y = 2 are values that satisfy our theorem.
The BIG IDEA of this theorem is that we can ALWAYS do the process above, no matter what
numbers are involved. There are definitely some difficulties in trying to do this in a general
way. Here are some:
• We won’t have specific numbers a and b to work with. We’ll have to describe the
process to make it work for any pair of integers with (a, b) = 1.
• We don’t even know how many steps of The Euclidean Algorithm that will have to
be used.
A way around the difficulties above is to use Proof by Induction on the number of steps
that The Euclidean Algorithm requires to do this. More specifically, we will show that for
any integers a and b with (a, b) = 1 that require only one step of The Euclidean Algorithm
have x and y satisfying the theorem. Then we will show that if we can always find x and y
when we must use k stages of Euclidean Algorithm then it shouldn’t be hard to find x and
y when must use k + 1 stages too!
Throughout we can assume that a > b > 0 since our version of The Euclidean Algorithm
tells us to replace (a, b) with (−a, −b), (−a, b), or (a, −b) whenever appropriate.
Proof. Let a and b be integers with (a, b) = 1 We will prove the existence of x and y with
ax + by = 1
by inducting on k, the number of stages used in The Euclidean Algorithm.
Assume that only one stage is required (k = 1). Then as in the proof of The Euclidean
Algorithm,
a = n1 b + r1 and (a, b) = 1 imply r1 =
and b =
.
So it is easy to find x and y with ax + by = 1 at this stage.
Next we assume our induction hypothesis, which must say something to the effect that using
The Euclidean Algorithm up to stage k still makes it possible to find x and y with
ax + by = 1. This can (and should) be written in a much better and more specific way!
Actual Induction Hypothesis:
Assume that stage k + 1 is required. Then as in the proof of The Euclidean Algorithm,
rk−1 = nk+1 rk + rk+1 and (a, b) = 1 imply rk+1 =
and rk =
.
Then . . .
Theorems 1.38 and Theorem 1.40 will soon be closed “closed" for
submission. After we do Theorem 1.40 you can do the following instead:
1. Find an online proof of Theorem 1.40 that uses The Well-Ordering Axiom.
2. Compare and contrast the proof given here (when (a,b)=1) versus the proof that you
found online. Discuss ways in which each proof could be improved for an audience
of students in a Number Theory course. Be sure to discuss specific aspects of the
mathematical arguements.
3. Your submission should typed, double-spaced, and at least a page but no more than
two pages. If you choose to include mathematical symbols in your submission, these
can be written by hand.
4. This assignment will be graded on a four point scale and is considered “open".