Lamé's Theorem

Theorem
Let $a, b \in \Z_{>0}$ be (strictly) positive integers.

Let $c$ and $d$ be the number of digits in $a$ and $b$ respectively when expressed in decimal notation.

Let the Euclidean Algorithm be employed to find the GCD of $a$ and $b$.

Then it will take fewer than $5 \times \min \set {c, d}$ integer divisions to find $\gcd \set {a, b}$.

Lemma
suppose $a \ge b$.

Then $\min \set {c, d}$ is the number of digits in $b$.

By Number of Digits in Number, we have:
 * $\min \set {c, d} = \floor {\log b} + 1$

it takes at least $5 \paren {\floor {\log b} + 1}$ cycles around the Euclidean Algorithm to find $\gcd \set {a, b}$.

Then we have:

For $b = 1$, both sides are equal to $1$, giving $1 > 1$, which is a contradiction.

Hence we consider $b > 1$ and take $\log$ on both sides:

However, $\dfrac 1 {\log \phi} \approx 4.785 < 5$.

This is a contradiction.

Hence the result by Proof by Contradiction.