Question Number 77459 by jagoll last updated on 06/Jan/20
$$ \\ $$ $$ \\ $$ $${the}\:{greatest}\:{value}\:{of}\:{n}\:{so}\:{that}\: \\ $$ $${only}\:{one}\:{value}\:{of}\:{k}\:{satisfies}\: \\ $$ $$\frac{\mathrm{8}}{\mathrm{15}}<\frac{{n}}{{n}+{k}}<\frac{\mathrm{7}}{\mathrm{13}}\:{is}\: \\ $$
Answered by MJS last updated on 06/Jan/20
$$\frac{\mathrm{8}}{\mathrm{15}}<\frac{{n}}{{n}+{k}}\:\Rightarrow\:{k}<\frac{\mathrm{7}}{\mathrm{8}}{n} \\ $$ $$\frac{{n}}{{n}+{k}}<\frac{\mathrm{7}}{\mathrm{13}}\:\Rightarrow\:{k}>\frac{\mathrm{6}}{\mathrm{7}}{n} \\ $$ $$\Rightarrow \\ $$ $$\frac{\mathrm{6}}{\mathrm{7}}{n}<{k}<\frac{\mathrm{7}}{\mathrm{8}}{n} \\ $$ $$\mathrm{to}\:\mathrm{get}\:\mathrm{at}\:\mathrm{least}\:{m}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{for}\:{k} \\ $$ $${n}\geqslant\mathrm{7}×\mathrm{8}×{m}+\mathrm{1} \\ $$ $$\Rightarrow\:\mathrm{answer}\:\mathrm{is}\:{n}=\mathrm{7}×\mathrm{8}×\mathrm{2}=\mathrm{112} \\ $$