Question Number 202109 by hardmath last updated on 20/Dec/23
$$ \\ $$How many different three-digit numbers a satisfy the condition GCD(a;18)>=2 ?
Answered by mr W last updated on 22/Dec/23
$$\mathrm{100}\leqslant{a}\leqslant\mathrm{999} \\ $$$$\mathrm{18}=\mathrm{2}×\mathrm{3}^{\mathrm{2}} \\ $$$$\mathrm{18}\:{has}\:{following}\:{factors}\:{which}\:{are} \\ $$$${equal}\:{or}\:{greater}\:{than}\:\mathrm{2}: \\ $$$$\mathrm{2},\:\mathrm{3},\:\mathrm{6},\:\mathrm{9},\:\mathrm{18} \\ $$$${that}\:{means}\:{a}\:{must}\:{be}\:{a}\:{multiple}\:{of} \\ $$$$\mathrm{2}\:{or}\:\mathrm{3}\:{or}\:\mathrm{6}\:{or}\:\mathrm{9}\:{or}\:\mathrm{18}. \\ $$$${a}=\mathrm{2}{k}:\:\Rightarrow\:\lfloor\frac{\mathrm{999}}{\mathrm{2}}\rfloor−\lfloor\frac{\mathrm{99}}{\mathrm{2}}\rfloor=\mathrm{499}−\mathrm{49}=\mathrm{450} \\ $$$${a}=\mathrm{3}{k}:\:\Rightarrow\:\lfloor\frac{\mathrm{999}}{\mathrm{3}}\rfloor−\lfloor\frac{\mathrm{99}}{\mathrm{3}}\rfloor=\mathrm{333}−\mathrm{33}=\mathrm{300} \\ $$$${a}=\mathrm{6}{k}:\:\Rightarrow\:\lfloor\frac{\mathrm{999}}{\mathrm{6}}\rfloor−\lfloor\frac{\mathrm{99}}{\mathrm{6}}\rfloor=\mathrm{166}−\mathrm{16}=\mathrm{150} \\ $$$${a}=\mathrm{9}{k}:\:\Rightarrow\:\lfloor\frac{\mathrm{999}}{\mathrm{9}}\rfloor−\lfloor\frac{\mathrm{99}}{\mathrm{9}}\rfloor=\mathrm{111}−\mathrm{11}=\mathrm{100} \\ $$$${a}=\mathrm{18}{k}:\:\Rightarrow\:\lfloor\frac{\mathrm{999}}{\mathrm{18}}\rfloor−\lfloor\frac{\mathrm{99}}{\mathrm{18}}\rfloor=\mathrm{55}−\mathrm{5}=\mathrm{50} \\ $$$$ \\ $$$$\mathrm{18}{k}\:\:\:\:\:\:\mathrm{9}{k}\:\:\:\:\:\:\mathrm{6}{k}\:\:\:\:\:\:\:\mathrm{3}{k}\:\:\:\:\:\:\mathrm{2}{k} \\ $$$$\:\:\mathrm{50}+\mathrm{100}+\mathrm{150}+\mathrm{300}+\mathrm{450} \\ $$$$\:\:\mathrm{50}+\cancel{\underset{\mathrm{50}} {\mathrm{100}}}+\cancel{\underset{\mathrm{100}} {\mathrm{150}}}+\cancel{\underset{\cancel{\underset{\cancel{\underset{\mathrm{100}} {\mathrm{200}}}} {\mathrm{250}}}} {\mathrm{300}}}+\cancel{\underset{\cancel{\underset{} {\mathrm{400}}}} {\mathrm{450}}} \\ $$$$\mathrm{50}+\mathrm{50}+\mathrm{100}+\mathrm{100}+\mathrm{300}=\mathrm{600}\:{numbers}\:\checkmark \\ $$
Commented by hardmath last updated on 21/Dec/23
$$\mathrm{perfect}\:\mathrm{dear}\:\mathrm{professor},\:\mathrm{thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much} \\ $$
Commented by hardmath last updated on 21/Dec/23
$$ \\ $$My dear professor, excuse me, what does the red part on the last line mean? And where did they come from?
Commented by mr W last updated on 22/Dec/23
$${if}\:{a}\:{number}\:{is}\:{divisible}\:{by}\:\mathrm{18},\:{then} \\ $$$${it}\:{is}\:{also}\:{divisible}\:{by}\:\mathrm{9},\:{by}\:\mathrm{2},\:{by}\:\mathrm{6},\:{by}\:\mathrm{3}. \\ $$$${similarly}\:{if}\:{a}\:{number}\:{is}\:{divisible} \\ $$$${by}\:\mathrm{9},\:{then}\:{it}\:{is}\:{also}\:{divisible}\:{by}\:\mathrm{3}. \\ $$$${and}\:{if}\:{it}\:{is}\:{divisible}\:{by}\:\mathrm{6},\:{it}\:{is}\:{also} \\ $$$${divisible}\:{by}\:\mathrm{3}\:{and}\:{by}\:\mathrm{2}. \\ $$$${so}\:{we}\:{must}\:{substract}\:{the}\:{numbers} \\ $$$${which}\:{are}\:{doubly}\:{counted}\:{as}\:{shown} \\ $$$${in}\:{the}\:{last}\:{line}. \\ $$
Commented by hardmath last updated on 22/Dec/23
$$\mathrm{Excellent}\:\mathrm{dear}\:\mathrm{pfofessor}, \\ $$$$\mathrm{thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{mych} \\ $$