
# 1.6: The Euclidean Algorithm


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In this section, we describe a systematic method that determines the greatest common divisor of two integers, due to Euclid and thus called the Euclidean algorithm.

Lemma $$\PageIndex{1}$$

If $$a,b,q,r\in\ZZ$$ and $$a=qb+r$$, then $$\gcd(a,b)=\gcd(r,b)$$.

Proof

Note that by Theorem 1.5.2, we have $$\gcd(bq+r,b)=\gcd(b,r)$$.

Now to the Euclidean algorithm in its general form, which basically states that the greatest common divisor of two integers is the last non-zero remainder of successive divisions.

Theorem $$\PageIndex{1}$$

Let $$a,b\in\NN$$ and assume $$a\geq b$$. Define $$r_0=a$$, $$r_1=b$$, $$s_0=1$$, $$s_1=0$$, $$t_0=0$$, and $$t_1=1$$. Then apply the division algorithm successively to obtain quotients and remainders $$q_j,r_j\in\NN$$ satisfying $$r_j=r_{j+1}q_{j+1}+r_{j+2}$$ and $$0\leq r_{j+2}<r_{j+1}$$ for all $$j=0,1,\dots,n-2$$ where $$n$$ is defined so that $$r_{n+1}=0$$. Along the way, also define $$s_{j+1}=s_{j-1}-q_{j+1}s_j$$ and $$t_{j+1}=t_{j-1}-q_{j+1}t_j$$. Then $$\gcd(a,b)=r_n= s_{n+1}a+t_{n+1}b$$.

Proof

By applying the division algorithm, we see that \begin{aligned} r_0&=q_1r_1+r_2 \ \ \ \ \ 0\leq r_2<r_1, \\ r_1&=q_2r_2+r_3 \ \ \ \ \ 0\leq r_3<r_2, \\ &.\\ &.\\ &.\\ r_{n-2}&=q_{n-1}r_{n-1}+r_{n} \ \ \ \ \ 0\leq r_{n}<r_{n-1}, \\ r_{n-1}&=q_nr_{n}.\end{aligned} Notice that, we will have a remainder of $$0$$ eventually since all the remainders are integers and every remainder in the next step is less than the remainder in the previous one. By Lemma 1.6.1, we see that $\gcd(a,b)=\gcd(b,r_2)=\gcd(r_2,r_3)=\dots=\gcd(r_n,0)=r_n.$

The full version of this theorem, with the $$s_j$$’s and $$t_j$$, is called the extended Euclidean Algorithm, while a simpler version without those coefficients is know as Euclidean Algorithm.

The attentive reader will have seen that We did not actually prove that the $$s_j$$’s and $$t_j$$’s can be used, as claimed, to write the $$\gcd$$ as a linear combination of $$a$$ and $$b$$. This proof is left as an exercise, below.

Example $$\PageIndex{1}$$

We will find the greatest common divisor of $$4147$$ and $$10672$$:
Note that \begin{aligned} 10672&=4147\cdot 2+2378,\\ 4147&=2378\cdot 1+1769,\\ 2378&=1769\cdot 1+609,\\ 1769&=609\cdot 2 +551,\\ 609&=551\cdot 1+58, \\ 551&=58\cdot 9+ 29,\\ 58&=29\cdot 2,\\\end{aligned} Hence $$\gcd(4147,10672)=29.$$

Exercise $$\PageIndex{1}$$

1. Use the Euclidean algorithm to find the greatest common divisor of $$412$$ and $$32$$ and express it in terms of the two integers.
2. Use the Euclidean algorithm to find the greatest common divisor of $$780$$ and $$150$$ and express it in terms of the two integers.
3. Find the greatest common divisor of $$70,98,108$$.
4. Let $$a,b\in\NN$$ be even. Prove that $$\gcd(a,b)=2\gcd(\dfrac{a}{2}, \dfrac{b}{2}).$$
5. Show that if $$a\in\NN$$ is even and $$b\in\NN$$ is odd, then $$\gcd(a,b)=\gcd(\dfrac{a}{2}, b)$$.
6. Prove the extended part of the Extended Euclidean Algorithm.