Saturday, March 30, 2019

Choosing two numbers $a,b,$ such that the Euclidean algorithm takes 10 steps

The question is to find $2$ integers $a$,$b$ $\in \mathbb{Z}$ for which when applying the Euclidean Algorithm for finding the $\gcd \left(a,b\right)$ precisely $10$ steps are required.
This is what I have done:
Let $\left(a,b\right)$ = $\left(427,264\right)$
The $10$ steps for the $\gcd \left(427,264 \right)$ are as follows:



$427=264 \cdot 1+163$



$264=163\cdot1+101 $




$163=101\cdot1+62 $



$101=62\cdot1+39 $



$62=39\cdot1+23 $



$39=23\cdot1+16 $



$23=16\cdot1+7 $




$16=7\cdot2+2 $



$7=2\cdot3+1 $



$2=1\cdot2+0$



I just wanna know if what I have done is right??? or if possible note the place I gone wrong??

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