Euclidean Algorithm/Demonstration

Example of use of Euclidean Algorithm

Using the Euclidean Algorithm, we can investigate in detail what happens when we apply the Division Theorem repeatedly to $a$ and $b$.

$\ds a$

$=$

$\ds q_1 b + r_1$

$\ds b$

$=$

$\ds q_2 r_1 + r_2$

$\ds r_1$

$=$

$\ds q_3 r_2 + r_3$

$\ds \cdots$

$\ds $

$\ds r_{n - 2}$

$=$

$\ds q_n r_{n - 1} + r_n$

$\ds r_{n - 1}$

$=$

$\ds q_{n + 1} r_n + 0$

From the Division Theorem, we know that the remainder is always strictly less than the divisor.

That is, in $a = q b + r$:

$0 \le r < \size b$

So we know that:

$b > r_1 > r_2 > \ldots > r_{n - 2} > r_{n - 1} > r_n > 0$

So the algorithm has to terminate.

$\blacksquare$

1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-2}$ Divisibility: Theorem $\text {2-2}$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 12$: Highest common factors and Euclid's algorithm
1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.3$ The Euclidean Algorithm
1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): $\text{A}.1$: Number theory