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Question Number 13154 by tawa tawa last updated on 15/May/17

$$\mathrm{Find}\mathrm{the}\mathrm{smallest}\mathrm{number}\mathrm{such}\mathrm{that}\mathrm{when}\mathrm{divided}\mathrm{by}18\mathrm{the}\mathrm{remainder}\mathrm{is}17,$$$$\mathrm{When}\mathrm{divided}\mathrm{by}20\mathrm{the}\mathrm{remainder}\mathrm{is}19.\mathrm{and}\mathrm{when}\mathrm{divided}\mathrm{by}24\mathrm{the}\mathrm{remainder}$$$$\mathrm{is}23.$$

Answered by mrW1 last updated on 15/May/17

$$N=18i+17=20j+19=24k+23$$$$20j+19=18j+18+2j+1=18i+17$$$$\Rightarrow 18(j-i)+2j+2=0$$$$\Rightarrow 9(j-i)+j+1=0$$$$\Rightarrow j+1=9i^{\prime }$$$$\Rightarrow j=9i^{\prime }-1$$$$$$$$20(9i^{\prime }-1)+19=24k+23$$$$20\times 9i^{\prime }=24(k+1)$$$$5\times 3i^{\prime }=2(k+1)$$$$\Rightarrow i^{\prime }=2n$$$$\Rightarrow k=15n-1$$$$$$$$\Rightarrow N=24(15n-1)+23=360n-1(n=1,2,3...)$$$$Thenumberis359or719or...$$

Commented by RasheedSindhi last updated on 15/May/17

$$\mathrm{The}\mathrm{smallest}\mathrm{number}\mathrm{is}\mathrm{required}$$$$\mathrm{So}\mathrm{I}\mathrm{think}\mathrm{the}\mathrm{answer}\mathrm{is}\mathrm{only}359.$$

Commented by tawa tawa last updated on 15/May/17

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

Answered by RasheedSindhi last updated on 15/May/17

$$\ast \mathrm{Common}\mathrm{diffefence}\mathrm{of}\mathrm{divisor}$$$$\mathrm{and}\mathrm{its}\mathrm{remainder}$$$$=18-17=20-19=24-23=1$$$$\ast \mathrm{lcm}\mathrm{of}\mathrm{divisors}18,20,24:$$$$18=2.3^2$$$$20=2^2.5$$$$24=2^3.3$$$$\mathrm{lcm}\mathrm{is}{2}^3.3^2.5=360$$$$$$$$\ast \mathrm{lcm}-(\mathrm{common}\mathrm{diff}.\mathrm{of}\mathrm{divisor}\mathrm{\&}\mathrm{remainder})\mathrm{u}$$$$=360-1=359$$$$$$

Commented by tawa tawa last updated on 15/May/17

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

Commented by mrW1 last updated on 16/May/17

$${That}^{\prime }ssmart!$$

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