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Let-N-be-the-greatest-multiple-of-36-all-of-whose-digits-are-even-and-no-two-of-whose-digits-are-the-same-Find-the-remainder-when-N-is-divided-by-1000-

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Question Number 111831 by Aina Samuel Temidayo last updated on 05/Sep/20
Let N be the greatest multiple of 36 all of whose digits are even and no two of whose digits are the same. Find the remainder when N is divided by 1000.
$$\mathrm{Let}\:\mathrm{N}\:\mathrm{be}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{36}\:\mathrm{all} \\ $$$$\mathrm{of}\:\mathrm{whose}\:\mathrm{digits}\:\mathrm{are}\:\mathrm{even}\:\mathrm{and}\:\mathrm{no}\:\mathrm{two}\:\mathrm{of} \\ $$$$\mathrm{whose}\:\mathrm{digits}\:\mathrm{are}\:\mathrm{the}\:\mathrm{same}.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{remainder}\:\mathrm{when}\:\mathrm{N}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{1000}. \\ $$
Answered by Rasheed.Sindhi last updated on 05/Sep/20
36=4×9 ^▶ A number is divisible by 36 if it is divisible by 4 & 9 ^▶ A number is divisible by 9 if sum of digits of the number is divisible by 9. ^▶ A number is divisible by 4 if the two digit number composed of one′s & tens is multiple of 4 N consists of distinct & even digits :0,2,4,6,8 Only possible combination of digits for which the formed number may be divisible by 9 is 0,4,6,8.(sum will be 0+4+6+8=18 which is divisible by 9. N is greatest so most significant digit must be 8 and next be 6 if possible. 8 6 _ _. The remaining digits are 0 & 4 Two 2-digit numbers are possible 04 & 40.Both numbers are fortunately divisible by 4.But for N(the greatest) 40 is only option. ^• So now the number is 8640. ^• The remainder on dividing by 1000 is obviously : 640
$$\mathrm{36}=\mathrm{4}×\mathrm{9} \\ $$$$\:^{\blacktriangleright} {A}\:{number}\:{is}\:{divisible}\:{by}\:\mathrm{36}\:{if}\:{it} \\ $$$$\:\:\:\:{is}\:{divisible}\:{by}\:\mathrm{4}\:\&\:\mathrm{9} \\ $$$$\:^{\blacktriangleright} {A}\:{number}\:{is}\:{divisible}\:{by}\:\mathrm{9}\:{if} \\ $$$$\:\:\:{sum}\:{of}\:{digits}\:{of}\:{the}\:{number}\:{is} \\ $$$$\:\:\:\:{divisible}\:{by}\:\mathrm{9}. \\ $$$$\:^{\blacktriangleright} {A}\:{number}\:{is}\:{divisible}\:{by}\:\mathrm{4}\:{if} \\ $$$$\:\:\:{the}\:{two}\:{digit}\:{number}\:{composed} \\ $$$$\:\:\:{of}\:{one}'{s}\:\&\:{tens}\:{is}\:{multiple}\:{of}\:\mathrm{4} \\ $$$$ \\ $$$$\:\:\:{N}\:{consists}\:{of}\:{distinct}\:\&\:{even}\: \\ $$$$\:\:\:\:{digits}\::\mathrm{0},\mathrm{2},\mathrm{4},\mathrm{6},\mathrm{8} \\ $$$$\:\:\:\:{Only}\:{possible}\:{combination}\:{of}\:{digits}\:{for} \\ $$$$\:\:\:{which}\:{the}\:{formed}\:{number}\:{may}\:{be} \\ $$$$\:\:\:\:{divisible}\:{by}\:\mathrm{9}\:{is}\:\mathrm{0},\mathrm{4},\mathrm{6},\mathrm{8}.\left({sum}\right. \\ $$$$\:\:\:\:\:{will}\:{be}\:\mathrm{0}+\mathrm{4}+\mathrm{6}+\mathrm{8}=\mathrm{18}\:{which}\:{is} \\ $$$$\:\:\:\:{divisible}\:{by}\:\mathrm{9}. \\ $$$$\:\:\:{N}\:{is}\:{greatest}\:{so}\:{most}\:{significant} \\ $$$$\:\:\:{digit}\:{must}\:{be}\:\mathrm{8}\:{and}\:{next}\:{be}\:\mathrm{6}\:{if} \\ $$$$\:\:\:\:{possible}. \\ $$$$\:\:\:\:\mathrm{8}\:\mathrm{6}\:\_\:\_. \\ $$$$\:\:\:\:\mathcal{T}{he}\:{remaining}\:{digits}\:{are}\:\mathrm{0}\:\&\:\mathrm{4} \\ $$$$\:\:\:\mathcal{T}{wo}\:\mathrm{2}-{digit}\:{numbers}\:{are}\:{possible} \\ $$$$\:\:\:\mathrm{04}\:\&\:\mathrm{40}.{Both}\:{numbers}\:{are} \\ $$$$\:\:\:{fortunately}\:{divisible}\:{by}\:\mathrm{4}.{But} \\ $$$$\:\:{for}\:{N}\left({the}\:{greatest}\right)\:\mathrm{40}\:{is}\:{only} \\ $$$$\:\:{option}. \\ $$$$\:\:^{\bullet} {So}\:{now}\:{the}\:{number}\:{is}\:\mathrm{8640}. \\ $$$$\:\:^{\bullet} \mathcal{T}{he}\:{remainder}\:{on}\:{dividing}\:{by} \\ $$$$\:\:\:\:\:\mathrm{1000}\:{is}\:{obviously}\:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{640} \\ $$
Commented by Aina Samuel Temidayo last updated on 05/Sep/20
Cool. Thanks.
$$\mathrm{Cool}.\:\mathrm{Thanks}. \\ $$

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