Not Continued Fractions

Stage: 4 and 5 Challenge Level: Challenge Level:1

(a) Find all positive integers $x$, $y$ and $z$ such that: $$x +\frac{1}{y + 1/z} = N = \frac{10}{7}$$
(b) Show that when $N=10/7$ is replaced by $N=8/5$ it is impossible to find positive integer values of $x$, $y$ and $z$ for which the finite continued fraction on the left hand side is equal to $N$. Find another fraction (rational number) $N$ for which the same is true.


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Published February 1998.