Question Number 124824 by ZiYangLee last updated on 06/Dec/20
$$\mathrm{How}\:\mathrm{many}\:\mathrm{positive}\:\mathrm{four}-\mathrm{digits}\:\mathrm{integers} \\ $$$${abcd}\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{following}\:\mathrm{conditions}: \\ $$$$\left(\mathrm{i}\right)\:{abcd}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{hy}\:\mathrm{7}; \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{When}\:\mathrm{the}\:\mathrm{first}\:\mathrm{and}\:\mathrm{the}\:\mathrm{last}\:\mathrm{digits} \\ $$$$\:\:\:\mathrm{are}\:\mathrm{interchanged},\:\mathrm{the}\:\mathrm{resulting}\:\mathrm{number} \\ $$$$\:\:\:{dbca}\:\mathrm{is}\:\mathrm{still}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{four}-\mathrm{digits}\:\mathrm{number} \\ $$$$\:\:\:\mathrm{that}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{7}. \\ $$
Commented by mr W last updated on 06/Dec/20
$${abcd}\:{must}\:{be}\:{different}\:{digits}? \\ $$
Commented by ZiYangLee last updated on 06/Dec/20
$${well}\:{as}\:{the}\:{original}\:{question}\:{does}\:{not} \\ $$$${mention}\:{it},\:{i}\:{think}\:{abcd}\:{can}\:{be}\:{same} \\ $$$${digits}\:{also}\:\left(?\right. \\ $$
Commented by mr W last updated on 06/Dec/20
$${then} \\ $$$$\left({i}\right) \\ $$$$\mathrm{1001},\mathrm{1008},\mathrm{1015},...,\mathrm{9996} \\ $$$$\Rightarrow\mathrm{1}+\frac{\mathrm{9996}−\mathrm{1001}}{\mathrm{7}}=\mathrm{1286}\:{numbers} \\ $$