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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

abcdef^(−) =?;{a,b,c,d,e,f}={1,2,3,4,5,6} 6∣abcdef^(−) ,5∣abcde^(−) ,4∣abcd^(−) ,3∣abc^(−) ,2∣ab^(−) ≪⇆⇄⇆⇄⇆⇄⇆⇄⇆⇄⇆⇄⇆≫ 6∣abcdef⇒3∣abcdef 3∣abcdef ∧ 3∣abc⇒3∣def ^• 3∣abc ∧ 3∣def 5∣abcde⇒e=5 ^• e=5 2∣ab⇒b=2,4,6 ^• b=2,4,6 6∣abcdef⇒f=2,4,6 ^• f=2,4,6 4∣abcd⇒4∣cd⇒d=2,4,6 ^• d=2,4,6 There′re two pieces of the number both divisibleby 3 ⇒ ^• 3∣(a+b+c) ∧ 3∣(d+e+f) Let′s make def piece first: e=5 f & d are even(2,4,6) and d+e+f is multiple of 3: f=2⇒The possible values for def: 452,652 both aren′t multiple of 3 ∴ f≠2 f=4, def=254,654. 3∤254; 3∣654 ∴ 654 is succesful candidate for def f=6,def=256,456. only 456 is multiple of 3 and hence succesful candidate. Let′s make other piece abc: Case1:def=654: ∵ 4∣cd⇒cd=16,36, 56(rejected b/c 5 has been already used) ∴ c=1,3 ∵ b is even and only 2 has been remaine unused from the given set. ∴ b=2 Possible candidates for abc are: 321 & 123 (thanks God, 3 divides both) abcdef=123654,321654 (when def=654) Case1:def=456: d=4⇒c=2,4,6(latter two are already used) ∴c=2 but then there remains no value for b which is also even. ∴ When def=456 there′s no value for abcdef(required number) So finally abcdef=123654,321654

$$\overline {\mathrm{abcdef}}=?;\left\{\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},\mathrm{e},\mathrm{f}\right\}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6}\right\} \\ $$$$\mathrm{6}\overline {\mid\mathrm{abcdef}},\mathrm{5}\mid\overline {\mathrm{abcde}},\mathrm{4}\mid\overline {\mathrm{abcd}},\mathrm{3}\mid\overline {\mathrm{abc}},\mathrm{2}\mid\overline {\mathrm{ab}} \\ $$$$\ll\leftrightarrows\rightleftarrows\leftrightarrows\rightleftarrows\leftrightarrows\rightleftarrows\leftrightarrows\rightleftarrows\leftrightarrows\rightleftarrows\leftrightarrows\rightleftarrows\leftrightarrows\gg \\ $$$$\mathrm{6}\mid\mathrm{abcdef}\Rightarrow\mathrm{3}\mid\mathrm{abcdef} \\ $$$$\mathrm{3}\mid\mathrm{abcdef}\:\wedge\:\mathrm{3}\mid\mathrm{abc}\Rightarrow\mathrm{3}\mid\mathrm{def} \\ $$$$\:^{\bullet} \:\mathrm{3}\mid\mathrm{abc}\:\wedge\:\mathrm{3}\mid\mathrm{def} \\ $$$$\mathrm{5}\mid\mathrm{abcde}\Rightarrow\mathrm{e}=\mathrm{5} \\ $$$$\:^{\bullet} \mathrm{e}=\mathrm{5} \\ $$$$\mathrm{2}\mid\mathrm{ab}\Rightarrow\mathrm{b}=\mathrm{2},\mathrm{4},\mathrm{6} \\ $$$$\:^{\bullet} \mathrm{b}=\mathrm{2},\mathrm{4},\mathrm{6} \\ $$$$\mathrm{6}\mid\mathrm{abcdef}\Rightarrow\mathrm{f}=\mathrm{2},\mathrm{4},\mathrm{6} \\ $$$$\:^{\bullet} \mathrm{f}=\mathrm{2},\mathrm{4},\mathrm{6} \\ $$$$\mathrm{4}\mid\mathrm{abcd}\Rightarrow\mathrm{4}\mid\mathrm{cd}\Rightarrow\mathrm{d}=\mathrm{2},\mathrm{4},\mathrm{6} \\ $$$$\:^{\bullet} \mathrm{d}=\mathrm{2},\mathrm{4},\mathrm{6} \\ $$$$\mathrm{There}'\mathrm{re}\:\mathrm{two}\:\mathrm{pieces}\:\mathrm{of}\:\mathrm{the}\:\mathrm{number} \\ $$$$\mathrm{both}\:\mathrm{divisibleby}\:\mathrm{3}\:\Rightarrow \\ $$$$\:^{\bullet} \mathrm{3}\mid\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)\:\wedge\:\mathrm{3}\mid\left(\mathrm{d}+\mathrm{e}+\mathrm{f}\right) \\ $$$$\underline{\mathrm{Let}'\mathrm{s}\:\mathrm{make}\:\boldsymbol{\mathrm{def}}\:\mathrm{piece}\:\mathrm{first}:} \\ $$$$\mathrm{e}=\mathrm{5}\:\mathrm{f}\:\&\:\mathrm{d}\:\mathrm{are}\:\mathrm{even}\left(\mathrm{2},\mathrm{4},\mathrm{6}\right)\:\mathrm{and}\:\mathrm{d}+\mathrm{e}+\mathrm{f} \\ $$$$\mathrm{is}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{3}: \\ $$$$\mathrm{f}=\mathrm{2}\Rightarrow\mathrm{The}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{for}\:\mathrm{def}: \\ $$$$\mathrm{452},\mathrm{652}\:\mathrm{both}\:\mathrm{aren}'\mathrm{t}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{3} \\ $$$$\therefore\:\mathrm{f}\neq\mathrm{2} \\ $$$$\mathrm{f}=\mathrm{4},\:\mathrm{def}=\mathrm{254},\mathrm{654}.\:\mathrm{3}\nmid\mathrm{254};\:\mathrm{3}\mid\mathrm{654} \\ $$$$\therefore\:\mathrm{654}\:\mathrm{is}\:\mathrm{succesful}\:\mathrm{candidate}\:\mathrm{for}\:\mathrm{def} \\ $$$$\mathrm{f}=\mathrm{6},\mathrm{def}=\mathrm{256},\mathrm{456}.\:\mathrm{only}\:\mathrm{456}\:\mathrm{is}\:\mathrm{multiple}\: \\ $$$$\mathrm{of}\:\mathrm{3}\:\mathrm{and}\:\mathrm{hence}\:\mathrm{succesful}\:\mathrm{candidate}. \\ $$$$\underline{\mathrm{Let}'\mathrm{s}\:\mathrm{make}\:\mathrm{other}\:\mathrm{piece}\:\boldsymbol{\mathrm{abc}}:} \\ $$$$\:\mathrm{Case1}:\underline{\mathrm{def}=\mathrm{654}:}\:\because\:\mathrm{4}\mid\mathrm{cd}\Rightarrow\mathrm{cd}=\mathrm{16},\mathrm{36}, \\ $$$$\mathrm{56}\left({rejected}\:{b}/{c}\:\mathrm{5}\:{has}\:{been}\:{already}\right. \\ $$$$\left.{used}\right) \\ $$$$\therefore\:\mathrm{c}=\mathrm{1},\mathrm{3} \\ $$$$\because\:\mathrm{b}\:\mathrm{is}\:\mathrm{even}\:\mathrm{and}\:\mathrm{only}\:\mathrm{2}\:\mathrm{has}\:\mathrm{been}\: \\ $$$$\mathrm{remaine}\:\mathrm{unused}\:\mathrm{from}\:\mathrm{the}\:\mathrm{given}\:\mathrm{set}. \\ $$$$\therefore\:\mathrm{b}=\mathrm{2} \\ $$$$\mathrm{Possible}\:\mathrm{candidates}\:\mathrm{for}\:\mathrm{abc}\:\mathrm{are}: \\ $$$$\mathrm{321}\:\&\:\mathrm{123}\:\left(\mathrm{thanks}\:\mathrm{God},\:\mathrm{3}\:\mathrm{divides}\:\mathrm{both}\right) \\ $$$$\mathrm{abcdef}=\mathrm{123654},\mathrm{321654}\:\left(\mathrm{when}\:\mathrm{def}=\mathrm{654}\right) \\ $$$$\:\mathrm{Case1}:\underline{\mathrm{def}=\mathrm{456}:}\:\mathrm{d}=\mathrm{4}\Rightarrow\mathrm{c}=\mathrm{2},\mathrm{4},\mathrm{6}\left({latter}\right. \\ $$$$\left.{two}\:{are}\:{already}\:{used}\right) \\ $$$$\therefore\mathrm{c}=\mathrm{2}\:\mathrm{but}\:\mathrm{then}\:\mathrm{there}\:\mathrm{remains}\:\mathrm{no} \\ $$$$\mathrm{value}\:\mathrm{for}\:\mathrm{b}\:\mathrm{which}\:\mathrm{is}\:\mathrm{also}\:\mathrm{even}. \\ $$$$\therefore\:\mathrm{When}\:\mathrm{def}=\mathrm{456}\:\mathrm{there}'\mathrm{s}\:\mathrm{no}\:\mathrm{value} \\ $$$$\mathrm{for}\:\mathrm{abcdef}\left(\mathrm{required}\:\mathrm{number}\right) \\ $$$$\:\:\:\:\:\:\mathrm{So}\:\mathrm{finally} \\ $$$$\:\:\:\:\:\mathrm{abcdef}=\mathrm{123654},\mathrm{321654} \\ $$

Commented by mathdanisur last updated on 16/Aug/21

Nice Ser, thank you

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

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