Compute $gcd\left(1714, 1814\right)$ using Euclidean Algorithm

Question

So I know the answer for this is $2$, but based on my own work, I can't get to that solution. I haven't done a gcd before where $b>a$. I thought I could just flip the numbers and use the same method but that didn't seem to work. Here's what I have so far, what I am doing wrong?

$$\begin{align} \mathrm{gcd}(1714, 1814) &= \mathrm{gcd}(1814, 1714) \\ \mathrm{gcd}(1814, 1714) &= (1714, 100)\\ &= (100, 14)\\ &= (14, 9)\\ &= (9, 5)\\ &= (5, 4)\\ &= (4, 1)\\ &= (1, 0)\\ &= 1\\ \end{align} $$

I basically tried using the Euclidian algorithm method where you keep doing long division into each number to get the remainder and continue with that process.

Answer 1

\begin{align} GCD(1814,1714)&=(1714,100)\\ &=(100,14)\\ &=(14,2)\\ \end{align} So, $2$ is the $GCD$

Answer 2

The answer is clearly wrong; both $1714$ and $1814$ are even, so $2$ divides both; the gcd is at least $2$.

In your solution, you really should write $\gcd(1814,1714)=\gcd(1714,100)=\cdots$ etc. The remainder when you divide $100$ by $14$ is $2$ ($100=7\times 14+2$) so $\gcd(100,14)=\gcd(14,2)$ etc.

Answer 3

When you reach (100,14) the next step gives (14,2) since $14 \times 7 = 98$ and $100 - 98 = 2$.

Answer 4

written as a continued fraction:

$$ \frac{ 1814 }{ 1714 } = 1 + \frac{ 100 }{ 1714 } $$ $$ \frac{ 1714 }{ 100 } = 17 + \frac{ 14 }{ 100 } $$ $$ \frac{ 100 }{ 14 } = 7 + \frac{ 2 }{ 14 } $$ $$ \frac{ 14 }{ 2 } = 7 + \frac{ 0 }{ 2 } $$ Simple continued fraction tableau:
$$ \begin{array}{cccccccccc} & & 1 & & 17 & & 7 & & 7 & \\ \frac{ 0 }{ 1 } & \frac{ 1 }{ 0 } & & \frac{ 1 }{ 1 } & & \frac{ 18 }{ 17 } & & \frac{ 127 }{ 120 } & & \frac{ 907 }{ 857 } \end{array} $$ $$ $$ $$ \begin{array}{ccc} \frac{ 1 }{ 0 } & \mbox{digit} & 1 \\ \frac{ 1 }{ 1 } & \mbox{digit} & 17 \\ \frac{ 18 }{ 17 } & \mbox{digit} & 7 \\ \frac{ 127 }{ 120 } & \mbox{digit} & 7 \\ \frac{ 907 }{ 857 } & \mbox{digit} & 0 \\ \end{array} $$

$$ 907 \cdot 120 - 857 \cdot 127 = 1 $$

$$ \gcd( 1814, 1714 ) = 2 $$
$$ 1814 \cdot 120 - 1714 \cdot 127 = 2 $$