I promised some of my tutorial students a demonstration of how the ‘highschool’ approach to Euclid’s algorithm can be reversed to give rise to the extended Euclidean algorithm – as opposed to the version in their lecture notes, which finds both gcd(a,b) and x,y such that ax+by=gcd(a,b) in one pass, at the price of some notational complexity. To do so, it seemed worth recapping some of the properties of divisibility that make Euclid’s algorithm tick, and give an application for its extended form. That ended up taking four pages, so I figured I’d post it here as well… you can read it behind the cut, or download the LaTeX-formatted pdf version.
The Euclidean Algorithm
Euclid’s algorithm recovers the greatest common divisor of two numbers a and b- that is, the largest x with the property that x|a and x|b (we write
x = gcd(a, b)). Thus, anything that is true of divisors of a and b in general is true of their g.c.d in particular.
To this end, recall that if x|a and x|b then x|a ± b. Further, if x|b then x|λb for any integer λ. Combining the two, we have that
x|a and x|b ⇒ x|(a ± λb) for all λ (1)
How does this help us? Since gcd(a, b) = gcd(b, a), we can assume w.l.o.g that a ≥ b. Then by the quotient-remainder theorem, we can write a as a = qb + r for some integers q, r with 0 ≤ r < b. Rearranging gives us r = a -qb, so by (1) we have
x|a and x|b ⇒ x|(a – qb) = r (2)
So in particular
gcd(a, b)|a and gcd(a, b)|b ⇒ gcd(a, b)|r (3)
So, to solve the question of finding gcd(a, b), we can instead solve the question of finding gcd(b, r). Notice that b ≤ a and r < b, so this second question is easier.
Repeating this process, the pairs of terms keep decreasing until eventually we have the question “find the g.c.d. of r and 0” for some r. But as anything divides 0, gcd(r, 0) = r. So there is always a final question; that question is easy to answer; and its answer is the same as the original g.c.d. we were looking for!
Suppose then that we want to find the g.c.d. of 51 and 36. We first appeal to the quotient-remainder theorem:
51 = 36 + 15
Thus any divisor of 51 and 36 (including the greatest) is also a divisor of 15. So we turn our attention to the easier problem of the g.c.d. of 36 and 15. Again by quotient-remainder
36 = 2 * 15 + 6
so it suffices to find gcd(15,6). Noting that
15 = 2 * 6 + 3
we discover that this is the same as the g.c.d. of 6 and 3. But
6 = 2 * 3 + 0
so thatâ€™s simply gcd(3,0)=3. Thus we have
gcd(51, 36) = gcd(36, 15) = gcd(15, 6) = gcd(6, 3) = gcd(3, 0) = 3
The Extended Euclidean Algorithm
Typically, one tabulates their progress through the algorithm more compactly; using the previous example we have
51 = 36 + 15 (4)
36 = 2 * 15 + 6 (5)
15 = 2 * 6 + 3 (6)
6 = 2 * 3 + 0 (7)
With such a representation, we can now tackle a related problem – given a, b, finding x, y such that
ax + by = gcd(a, b)
For our example, this means finding x, y with the property that 51x+36y = 3.
Notice that we do not have an identity in terms of 3, 36 and 51 anywhere in the table. But from (6), we do have an identity for 3 in terms of 15 and 6:
3 = 15 – 2 * 6
Then (5) tells us that
6 = 36 – 2 * 15
so we can rewrite our result for 3 in terms of 36 and 15 instead of 15 and 6, and so on.
In this way, we can chain upwards through the table to an identity for 3 ultimately in terms of 51 and 36- this process is known as the extended euclidean algorithm. Whereas the Euclidean algorithm works down from a and b, through simpler and simpler terms to the g.c.d., so the extended version works up from that g.c.d. through increasingly complicated terms to an expression in terms of a and b.
For our example, this works as follows:
3 = 15 – 2 * 6 by (6)
= 15 – 2 * (36 – 2 * 15) by (5)
= 5 * 15 – 2 * 36 (simplifying)
= 5 * (51 – 36) – 2 * 36 by (4)
= 5 * 51 – 7 * 36 (simplifying)
Linear Diophantine Equations
Why is all this useful? In number theory, the study of Diophantine equations concerns finding integer solutions to equations, where possible. Typically we seek to classify such equations based on whether we can find infinitely many such solutions, only finitely many, or none at all. The general question as to whether solutions exist is actually undecidable, but in various special cases we can say something.
Armed with the extended euclidean algorithm, we can tackle the case of linear diophantine equations in two variables:
For fixed integers a, b, c, do there exist integer solutions x, y of ax + by = c ?
Suppose c = λgcd(a, b) for some integer λ . Then, by the extended Euclidean algorithm we have x’, y’ with the property axâ€™+byâ€™=gcd(a, b). So x = λx’, y = λy’ solve the linear diophantine equation, since
ax + by = aλx’ + bλy’ = λ(ax’ + by’) = λgcd(a, b) = c
Conversely, suppose c is not a multiple of gcd(a, b). Then, if the equation had solutions x, y then weâ€™d have that gcd(a, b)|a|ax and gcd(a, b)|b|by so gcd(a, b)|ax + by = c, which is a contradiction.
Hence, the linear diophantine equation in two variables ax + by = c has integer solutions if and only if c is a multiple of gcd(a, b).
Number of Solutions
Assuming solutions exist, how many are there? Although the g.c.d. is unique, suitable pairs (x, y) with ax+by = gcd(a, b) (and hence solutions to the linear diophantine equation) are not. To see this, consider one of the standard tricks of analysis, adding and subtracting the same thing:
ax+by = λgcd(a, b) ) ax+by+ab-ab = λgcd(a, b) ) a(x+b)+b(y-a) = λgcd(a, b)
That is, if (x, y) satisfy the equation, so do (x + b, y – a). Which means we can apply the result again, giving (x+2b, y-2a). These will always be pairs of integer solutions, and so the existence of just one pair (x, y) gives us an infinite family (x + ib, y – ia) of solutions.
Hence we conclude that the linear diophantine equation in two variables ax+by = c has infinitely many integer solutions if c is a multiple of gcd(a, b), and none otherwise.