Question Number 193892 by Mastermind last updated on 22/Jun/23
$$\mathrm{Ques}. \\ $$$$\:\:\:\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{integers}\:\mathrm{in}\:\mathrm{the}\:\mathrm{set} \\ $$$$\mathrm{S}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},…,\mathrm{60}\right\}\:\mathrm{which}\:\mathrm{are}\:\mathrm{not}\:\mathrm{divisible} \\ $$$$\mathrm{by}\:\mathrm{2}\:\mathrm{nor}\:\mathrm{by}\:\mathrm{3}\:\mathrm{nor}\:\mathrm{by}\:\mathrm{5}. \\ $$$$ \\ $$$$\mathrm{Hello} \\ $$
Answered by BaliramKumar last updated on 22/Jun/23
$$\mathrm{60}×\frac{\mathrm{2}−\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{3}−\mathrm{1}}{\mathrm{3}}×\frac{\mathrm{5}−\mathrm{1}}{\mathrm{5}}\: \\ $$$$\:\mathrm{60}×\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{2}}{\mathrm{3}}×\frac{\mathrm{4}}{\mathrm{5}}\:=\:\mathrm{16} \\ $$$$ \\ $$
Commented by SaRahAli last updated on 22/Jun/23
$$? \\ $$
Commented by Mastermind last updated on 22/Jun/23
$$\mathrm{Than}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much} \\ $$
Answered by MM42 last updated on 22/Jun/23
$${let}\:\:{A},{B},{C}\:\subset{S}\:\&\:{A}:\mathrm{2}{k}\:;\:{B}:\mathrm{3}{k}\:;\:{C}:\mathrm{5}{k} \\ $$$$\mid{A}\mid=\mathrm{30}\:\&\:\mid{B}\mid=\mathrm{20}\:\:\&\:\mid{C}\mid=\mathrm{12} \\ $$$$\mid{A}\cap{B}\mid=\mathrm{10}\:\&\:\mid{A}\cap{C}\mid=\mathrm{6}\:\&\:\mid{B}\cap{C}\mid=\mathrm{4}\:\&\:\mid{A}\cap{B}\cap{C}\mid=\mathrm{2} \\ $$$$\mid{A}\cup{B}\cup{C}\mid=\mathrm{30}+\mathrm{20}+\mathrm{12}−\mathrm{10}−\mathrm{6}−\mathrm{4}+\mathrm{2}=\mathrm{44} \\ $$$${ans}=\mathrm{60}−\mathrm{44}=\mathrm{16} \\ $$
Commented by Mastermind last updated on 22/Jun/23
$$\mathrm{Thank}\:\mathrm{you}\: \\ $$