Competition problem — Number theory

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Number theory Difficulty 6.6 National olympiad Prove it

5. Let d be a integer, $d \geqslant 2$ .
a) Show that the simple continued fraction of $\sqrt{d^{2}-1}$ is $[d-1 ; \overline{1,2 d-2}]$ .
b) Show that the simple continued fraction of $\sqrt{d^{2}-d}$ is $[d-1 ; \overline{2,2 d-2}]$ .
c) Use parts (a) and (b) to find the simple continued fractions of $\sqrt{99}, \sqrt{110}$ , $\sqrt{272}$ , and $\sqrt{600}$

Solution

5. с) $[9 ; \overline{1,18}],[10 ; \overline{2,20}],[16 ; \overline{2,32}],[24 ; \overline{2,48}]$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.