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Question Number 150884 by mathdanisur last updated on 16/Aug/21

Answered by Rasheed.Sindhi last updated on 16/Aug/21

$$\overline{\mathrm{abcdef}}=?;\{\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},\mathrm{e},\mathrm{f}\}=\{1,2,3,4,5,6\}$$$$6\overline{\mid \mathrm{abcdef}},5\mid \overline{\mathrm{abcde}},4\mid \overline{\mathrm{abcd}},3\mid \overline{\mathrm{abc}},2\mid \overline{\mathrm{ab}}$$$$\ll \leftrightarrows \rightleftarrows \leftrightarrows \rightleftarrows \leftrightarrows \rightleftarrows \leftrightarrows \rightleftarrows \leftrightarrows \rightleftarrows \leftrightarrows \rightleftarrows \leftrightarrows \gg$$$$6\mid \mathrm{abcdef}\Rightarrow 3\mid \mathrm{abcdef}$$$$3\mid \mathrm{abcdef}\wedge 3\mid \mathrm{abc}\Rightarrow 3\mid \mathrm{def}$$$${}^{\bullet }3\mid \mathrm{abc}\wedge 3\mid \mathrm{def}$$$$5\mid \mathrm{abcde}\Rightarrow \mathrm{e}=5$$$${}^{\bullet }\mathrm{e}=5$$$$2\mid \mathrm{ab}\Rightarrow \mathrm{b}=2,4,6$$$${}^{\bullet }\mathrm{b}=2,4,6$$$$6\mid \mathrm{abcdef}\Rightarrow \mathrm{f}=2,4,6$$$${}^{\bullet }\mathrm{f}=2,4,6$$$$4\mid \mathrm{abcd}\Rightarrow 4\mid \mathrm{cd}\Rightarrow \mathrm{d}=2,4,6$$$${}^{\bullet }\mathrm{d}=2,4,6$$$${\mathrm{There}}^{\prime }\mathrm{re}\mathrm{two}\mathrm{pieces}\mathrm{of}\mathrm{the}\mathrm{number}$$$$\mathrm{both}\mathrm{divisibleby}3\Rightarrow$$$${}^{\bullet }3\mid (\mathrm{a}+\mathrm{b}+\mathrm{c})\wedge 3\mid (\mathrm{d}+\mathrm{e}+\mathrm{f})$$$$\underset{―}{{\mathrm{Let}}^{\prime }\mathrm{s}\mathrm{make}\mathbf{def}\mathrm{piece}\mathrm{first}:}$$$$\mathrm{e}=5\mathrm{f}\mathrm{\&}\mathrm{d}\mathrm{are}\mathrm{even}(2,4,6)\mathrm{and}\mathrm{d}+\mathrm{e}+\mathrm{f}$$$$\mathrm{is}\mathrm{multiple}\mathrm{of}3:$$$$\mathrm{f}=2\Rightarrow \mathrm{The}\mathrm{possible}\mathrm{values}\mathrm{for}\mathrm{def}:$$$$452,652\mathrm{both}{\mathrm{aren}}^{\prime }\mathrm{t}\mathrm{multiple}\mathrm{of}3$$$$\therefore \mathrm{f}\ne 2$$$$\mathrm{f}=4,\mathrm{def}=254,654.3\nmid 254;3\mid 654$$$$\therefore 654\mathrm{is}\mathrm{succesful}\mathrm{candidate}\mathrm{for}\mathrm{def}$$$$\mathrm{f}=6,\mathrm{def}=256,456.\mathrm{only}456\mathrm{is}\mathrm{multiple}$$$$\mathrm{of}3\mathrm{and}\mathrm{hence}\mathrm{succesful}\mathrm{candidate}.$$$$\underset{―}{{\mathrm{Let}}^{\prime }\mathrm{s}\mathrm{make}\mathrm{other}\mathrm{piece}\mathbf{abc}:}$$$$\mathrm{Case}1:\underset{―}{\mathrm{def}=654:}\because 4\mid \mathrm{cd}\Rightarrow \mathrm{cd}=16,36,$$$$56(rejectedb/c5hasbeenalready$$$$used)$$$$\therefore \mathrm{c}=1,3$$$$\because \mathrm{b}\mathrm{is}\mathrm{even}\mathrm{and}\mathrm{only}2\mathrm{has}\mathrm{been}$$$$\mathrm{remaine}\mathrm{unused}\mathrm{from}\mathrm{the}\mathrm{given}\mathrm{set}.$$$$\therefore \mathrm{b}=2$$$$\mathrm{Possible}\mathrm{candidates}\mathrm{for}\mathrm{abc}\mathrm{are}:$$$$321\mathrm{\&}123(\mathrm{thanks}\mathrm{God},3\mathrm{divides}\mathrm{both})$$$$\mathrm{abcdef}=123654,321654(\mathrm{when}\mathrm{def}=654)$$$$\mathrm{Case}1:\underset{―}{\mathrm{def}=456:}\mathrm{d}=4\Rightarrow \mathrm{c}=2,4,6(latter$$$$twoarealreadyused)$$$$\therefore \mathrm{c}=2\mathrm{but}\mathrm{then}\mathrm{there}\mathrm{remains}\mathrm{no}$$$$\mathrm{value}\mathrm{for}\mathrm{b}\mathrm{which}\mathrm{is}\mathrm{also}\mathrm{even}.$$$$\therefore \mathrm{When}\mathrm{def}=456{\mathrm{there}}^{\prime }\mathrm{s}\mathrm{no}\mathrm{value}$$$$\mathrm{for}\mathrm{abcdef}\left(\mathrm{required}\mathrm{number}\right)$$$$\mathrm{So}\mathrm{finally}$$$$\mathrm{abcdef}=123654,321654$$

Commented by mathdanisur last updated on 16/Aug/21

$$\mathrm{Nice}\mathrm{Ser},\mathrm{thank}\mathrm{you}$$

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