How-many-positive-four-digits-integers-abcd-satisfy-the-following-conditions-i-abcd-is-divisible-hy-7-ii-When-the-first-and-the-last-digits-are-interchanged-the-resulting-number-dbca-is-s

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Question Number 124824 by ZiYangLee last updated on 06/Dec/20

$$\mathrm{How}\mathrm{many}\mathrm{positive}\mathrm{four}-\mathrm{digits}\mathrm{integers}$$$$abcd\mathrm{satisfy}\mathrm{the}\mathrm{following}\mathrm{conditions}:$$$$\left(\mathrm{i}\right)abcd\mathrm{is}\mathrm{divisible}\mathrm{hy}7;$$$$\left(\mathrm{ii}\right)\mathrm{When}\mathrm{the}\mathrm{first}\mathrm{and}\mathrm{the}\mathrm{last}\mathrm{digits}$$$$\mathrm{are}\mathrm{interchanged},\mathrm{the}\mathrm{resulting}\mathrm{number}$$$$dbca\mathrm{is}\mathrm{still}\mathrm{a}\mathrm{positive}\mathrm{four}-\mathrm{digits}\mathrm{number}$$$$\mathrm{that}\mathrm{is}\mathrm{divisible}\mathrm{by}7.$$

Commented by mr W last updated on 06/Dec/20

$$abcdmustbedifferentdigits?$$

Commented by ZiYangLee last updated on 06/Dec/20

$$wellastheoriginalquestiondoesnot$$$$mentionit,ithinkabcdcanbesame$$$$digitsalso(?$$

Commented by mr W last updated on 06/Dec/20

$$then$$$$\left(i\right)$$$$1001,1008,1015,\dots ,9996$$$$\Rightarrow 1+\frac{9996-1001}{7}=1286numbers$$

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