Question Number 149323 by naka3546 last updated on 04/Aug/21
$${How}\:\:{many}\:\:{ordered}\:\:{pairs}\:\left({a},{b}\right)\:{with}\:\:{b}\:<\:{a}\:<\:\mathrm{100}\:\:,\:\:{a},{b}\:\in\:\mathbb{N}\:\:{such}\:\:{that}\:\:\frac{{a}}{{b}}\:\:{both}\:\:\:\frac{{a}+\mathrm{1}}{{b}+\mathrm{1}}\:\:{are}\:\:{integers}\:\:.\: \\ $$
Answered by Olaf_Thorendsen last updated on 05/Aug/21
$$\mathrm{N}\:=\:\underset{{b}=\mathrm{1}} {\overset{\mathrm{98}} {\sum}}\underset{{a}={b}+\mathrm{1}} {\overset{\mathrm{99}} {\sum}}\lfloor\frac{\lfloor\frac{{a}}{{b}}\rfloor}{\frac{{a}}{{b}}}×\frac{\lfloor\frac{{a}+\mathrm{1}}{{b}+\mathrm{1}}\rfloor}{\frac{{a}+\mathrm{1}}{{b}+\mathrm{1}}}\rfloor \\ $$$$\mathrm{N}\:=\:\mathrm{85} \\ $$
Commented by naka3546 last updated on 05/Aug/21
$${thank}\:{you},\:{sir}. \\ $$
Answered by Olaf_Thorendsen last updated on 05/Aug/21
