Question Number 99058 by Learner-123 last updated on 18/Jun/20
$${Find}\:{a}\:{number}\:{that}\:{becomes}\:{N}\:{times} \\ $$$${smaller}\:{if}\:{the}\:{first}\:{digit}\:{is}\:{removed} \\ $$$${in}\:{the}\:{following}\:{cases}: \\ $$$$\left.\mathrm{1}\right)\:{N}=\mathrm{17} \\ $$$$\left.\mathrm{2}\right)\:{N}=\mathrm{27} \\ $$$$\left.\mathrm{3}\right){N}=\mathrm{37} \\ $$$$\left.\mathrm{4}\right){N}=\mathrm{47}. \\ $$
Commented by Rasheed.Sindhi last updated on 18/Jun/20
$$\left.\mathrm{1}\right)\:{N}=\mathrm{17} \\ $$
Commented by mr W last updated on 18/Jun/20
$${i}\:{don}'{t}\:{understand}\:{what}\:{is}\:{meant}. \\ $$$${say}\:{the}\:{number}\:{is}\:{XYYYYY}. \\ $$$${it}\:{should}\:{be} \\ $$$${YYYYY}=\frac{{XYYYYY}}{{N}}\:{with}\:{N}=\mathrm{17}? \\ $$
Commented by Rasheed.Sindhi last updated on 18/Jun/20
$${Sir}\:{I}\:{understand}\:{by}\:{first}\:{digit} \\ $$$$'{unit}-{digit}'.{So}\:{if}\:{remove}\:\mathrm{7}\:{from} \\ $$$$\mathrm{17}\:{it}\:{becomes}\:\mathrm{1}{which}\:{is}\:\mathrm{17}\:{times} \\ $$$${smaller}\:{than}\:\mathrm{17}. \\ $$
Commented by mr W last updated on 18/Jun/20
$${thanks}\:{sir}! \\ $$$$\left({Xy}\right)=\mathrm{10}{X}+{y}=\mathrm{17}{X} \\ $$$$\Rightarrow{y}=\mathrm{7}{X}\leqslant\mathrm{9} \\ $$$$\Rightarrow{X}\leqslant\frac{\mathrm{9}}{\mathrm{7}}\:\Rightarrow{X}=\mathrm{1}\:\Rightarrow{y}=\mathrm{7} \\ $$$$\Rightarrow{the}\:{number}\:{is}\:\mathrm{17}. \\ $$$$ \\ $$$$\left({Xy}\right)=\mathrm{10}{X}+{y}=\mathrm{27}{X} \\ $$$$\Rightarrow{y}=\mathrm{17}{X}\leqslant\mathrm{9}\:\Rightarrow{X}\leqslant\frac{\mathrm{9}}{\mathrm{17}}\:\Rightarrow{no}\:{solution} \\ $$
Commented by Learner-123 last updated on 18/Jun/20
$${thanks}\:{both}\:{sirs}! \\ $$
Commented by Rasheed.Sindhi last updated on 18/Jun/20
$${First}-{digit}\left({unit}-{digit}\right){removing}: \\ $$$${tu}=\mathrm{10}{t}+{u}\rightarrow\frac{\left(\mathrm{10}{t}+{u}\right)−{u}}{\mathrm{10}}={u} \\ $$$$\frac{\left(\mathrm{10}{t}+{u}\right)−{u}}{\mathrm{10}}=\frac{\mathrm{10}{t}+{u}}{\mathrm{10}{t}+{u}}=\mathrm{1} \\ $$$${t}=\mathrm{1} \\ $$$${The}\:{required}\:{number}\:{is}\:\mathrm{1}{u} \\ $$$$\mathrm{11},\mathrm{12},\mathrm{13},…,\mathrm{19} \\ $$