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Question Number 17401 by ajfour last updated on 05/Jul/17

Find the sum of 4-digit greatest number and the 5-digit smallest number, each number having three different digits.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{4}-\mathrm{digit}\:\mathrm{greatest} \\ $$$$\mathrm{number}\:\mathrm{and}\:\mathrm{the}\:\mathrm{5}-\mathrm{digit}\:\mathrm{smallest} \\ $$$$\mathrm{number},\:\mathrm{each}\:\mathrm{number}\:\mathrm{having}\:\mathrm{three} \\ $$$$\mathrm{different}\:\mathrm{digits}. \\ $$

Commented by RasheedSoomro last updated on 05/Jul/17

Yes sir you are very right! I am going to correct.

$$\mathrm{Yes}\:\mathrm{sir}\:\mathrm{you}\:\mathrm{are}\:\boldsymbol{\mathrm{very}}\:\mathrm{right}! \\ $$$$\mathrm{I}\:\mathrm{am}\:\mathrm{going}\:\mathrm{to}\:\mathrm{correct}. \\ $$

Commented by RasheedSoomro last updated on 05/Jul/17

4-digit greatest number: 9999 Greatest number with required condition: 9987 process: Changing at least two digits can make 9999 with three different digits. we select last two digits for changing to keep the number near to 9999. ten^(th) - place should be decreased at least. ⇒9989 unit place also should be decreased at least but also the digitshould be different from 8 ⇒99987 −−−−−−−−−−− 5-digit smallest number 10000 has already two different digits, it requires change in only one digit.This change should be made in least significant digit i-e unit digit.Also this change should be as small as possible and other than 1 So the 5-digit smallest with three different digits is 10002 Sum of such numbers 19989

$$\mathrm{4}-\mathrm{digit}\:\mathrm{greatest}\:\mathrm{number}:\:\mathrm{9999} \\ $$$$\mathrm{Greatest}\:\mathrm{number}\:\mathrm{with}\:\mathrm{required} \\ $$$$\mathrm{condition}:\:\mathrm{9987} \\ $$$$\mathrm{process}: \\ $$$$\mathrm{Changing}\:\mathrm{at}\:\mathrm{least}\:\mathrm{two}\:\mathrm{digits}\:\mathrm{can}\:\mathrm{make} \\ $$$$\mathrm{9999}\:\mathrm{with}\:\mathrm{three}\:\mathrm{different}\:\mathrm{digits}. \\ $$$$\mathrm{we}\:\mathrm{select}\:\mathrm{last}\:\mathrm{two}\:\mathrm{digits}\:\mathrm{for}\:\mathrm{changing} \\ $$$$\mathrm{to}\:\mathrm{keep}\:\mathrm{the}\:\mathrm{number}\:\mathrm{near}\:\mathrm{to}\:\mathrm{9999}. \\ $$$$\mathrm{ten}^{\mathrm{th}} -\:\mathrm{place}\:\mathrm{should}\:\mathrm{be}\:\mathrm{decreased}\:\boldsymbol{\mathrm{at}}\:\boldsymbol{\mathrm{least}}. \\ $$$$\Rightarrow\mathrm{9989} \\ $$$$\mathrm{unit}\:\mathrm{place}\:\mathrm{also}\:\:\mathrm{should}\:\mathrm{be}\:\mathrm{decreased}\:\mathrm{at}\:\mathrm{least} \\ $$$$\mathrm{but}\:\mathrm{also}\:\mathrm{the}\:\mathrm{digitshould}\:\mathrm{be}\:\mathrm{different}\:\mathrm{from}\:\mathrm{8}\:\Rightarrow\mathrm{99987} \\ $$$$−−−−−−−−−−− \\ $$$$\mathrm{5}-\mathrm{digit}\:\mathrm{smallest}\:\mathrm{number}\:\mathrm{10000}\:\mathrm{has} \\ $$$$\mathrm{already}\:\mathrm{two}\:\mathrm{different}\:\mathrm{digits},\:\mathrm{it}\:\mathrm{requires} \\ $$$$\mathrm{change}\:\mathrm{in}\:\mathrm{only}\:\mathrm{one}\:\mathrm{digit}.\mathrm{This}\:\mathrm{change} \\ $$$$\mathrm{should}\:\mathrm{be}\:\mathrm{made}\:\mathrm{in}\:\mathrm{least}\:\mathrm{significant}\:\mathrm{digit} \\ $$$$\mathrm{i}-\mathrm{e}\:\mathrm{unit}\:\mathrm{digit}.\mathrm{Also}\:\mathrm{this}\:\mathrm{change}\:\mathrm{should}\:\mathrm{be} \\ $$$$\mathrm{as}\:\mathrm{small}\:\mathrm{as}\:\mathrm{possible}\:\mathrm{and}\:\mathrm{other}\:\mathrm{than}\:\mathrm{1} \\ $$$$\mathrm{So}\:\mathrm{the}\:\mathrm{5}-\mathrm{digit}\:\mathrm{smallest}\:\mathrm{with}\:\mathrm{three}\:\mathrm{different} \\ $$$$\mathrm{digits}\:\mathrm{is}\:\mathrm{10002} \\ $$$$\mathrm{Sum}\:\mathrm{of}\:\mathrm{such}\:\mathrm{numbers}\:\mathrm{19989} \\ $$

Commented by student last updated on 05/Jul/17

Yes, it should be 10002.

$$\mathrm{Yes},\:\mathrm{it}\:\mathrm{should}\:\mathrm{be}\:\mathrm{10002}. \\ $$

Commented by mrW1 last updated on 05/Jul/17

why 10012, not 10002?

$$\mathrm{why}\:\mathrm{10012},\:\mathrm{not}\:\mathrm{10002}? \\ $$

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