Competition problem — Number theory

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Number theory Difficulty 6.3 National olympiad Prove it

3. Under the conditions and notations of Question 2, prove:
(i) $\left|v_{h+1}\right|=\left(v_{0}, v_{1}\right)$ ;
(ii) $d \mid v_{0}$ and $d \mid v_{1}$ if and only if $d \mid v_{h+1}$ ;
(iii) there exist integers $x_{0}, x_{1}$ , such that $v_{h+1}=x_{0} v_{0}+x_{1} v_{1}$ .

Solution

None

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