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Let-Akbar-and-Birbal-together-have-n-marbles-where-n-gt-0-Akbar-says-to-Birbal-If-I-give-you-some-marbles-then-you-will-have-twice-as-many-marbles-as-I-will-have-Birbal-says-to-Akbar-If-I-gi

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Question Number 19516 by Tinkutara last updated on 12/Aug/17
Let Akbar and Birbal together have n marbles, where n > 0. Akbar says to Birbal, “If I give you some marbles then you will have twice as many marbles as I will have.” Birbal says to Akbar, “If I give you some marbles then you will have thrice as many marbles as I will have.” What is the minimum possible value of n for which the above statements are true?
$$\mathrm{Let}\:\mathrm{Akbar}\:\mathrm{and}\:\mathrm{Birbal}\:\mathrm{together}\:\mathrm{have}\:{n} \\ $$$$\mathrm{marbles},\:\mathrm{where}\:{n}\:>\:\mathrm{0}. \\ $$$$\mathrm{Akbar}\:\mathrm{says}\:\mathrm{to}\:\mathrm{Birbal},\:“\mathrm{If}\:\mathrm{I}\:\mathrm{give}\:\mathrm{you}\:\mathrm{some} \\ $$$$\mathrm{marbles}\:\mathrm{then}\:\mathrm{you}\:\mathrm{will}\:\mathrm{have}\:\mathrm{twice}\:\mathrm{as} \\ $$$$\mathrm{many}\:\mathrm{marbles}\:\mathrm{as}\:\mathrm{I}\:\mathrm{will}\:\mathrm{have}.''\:\mathrm{Birbal} \\ $$$$\mathrm{says}\:\mathrm{to}\:\mathrm{Akbar},\:“\mathrm{If}\:\mathrm{I}\:\mathrm{give}\:\mathrm{you}\:\mathrm{some} \\ $$$$\mathrm{marbles}\:\mathrm{then}\:\mathrm{you}\:\mathrm{will}\:\mathrm{have}\:\mathrm{thrice}\:\mathrm{as} \\ $$$$\mathrm{many}\:\mathrm{marbles}\:\mathrm{as}\:\mathrm{I}\:\mathrm{will}\:\mathrm{have}.'' \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of} \\ $$$${n}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{above}\:\mathrm{statements}\:\mathrm{are} \\ $$$$\mathrm{true}? \\ $$
Answered by dioph last updated on 12/Aug/17
As n ∈ N, if there is the possibility B = 2x_1 and A = x_1 ⇒ n = 3x_1 . Similarly, when A = 3x_2 and B = x_2 ⇒ n = 4x_2 . Hence 3 ∣ n and 4 ∣ n ⇒ n ≥ 12. If n = 12, 4<A<9, and B = 12−A, then we have a solution where A can give marbles so that A = 4 and B = 8, and B can give marbles so that A = 9 and B = 3.
$$\mathrm{As}\:{n}\:\in\:\mathbb{N},\:\mathrm{if}\:\mathrm{there}\:\mathrm{is}\:\mathrm{the}\:\mathrm{possibility}\: \\ $$$$\mathrm{B}\:=\:\mathrm{2}{x}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{A}\:=\:{x}_{\mathrm{1}} \:\Rightarrow\:{n}\:=\:\mathrm{3}{x}_{\mathrm{1}} . \\ $$$$\mathrm{Similarly},\:\mathrm{when}\:{A}\:=\:\mathrm{3}{x}_{\mathrm{2}} \:\mathrm{and}\:{B}\:=\:{x}_{\mathrm{2}} \\ $$$$\Rightarrow\:{n}\:=\:\mathrm{4}{x}_{\mathrm{2}} .\:\mathrm{Hence}\:\mathrm{3}\:\mid\:{n}\:\mathrm{and}\:\mathrm{4}\:\mid\:{n} \\ $$$$\Rightarrow\:{n}\:\geqslant\:\mathrm{12}.\:\mathrm{If}\:{n}\:=\:\mathrm{12},\:\mathrm{4}<{A}<\mathrm{9}, \\ $$$$\mathrm{and}\:{B}\:=\:\mathrm{12}−{A},\:\mathrm{then}\:\mathrm{we}\:\mathrm{have}\:\mathrm{a}\:\mathrm{solution} \\ $$$$\mathrm{where}\:{A}\:\mathrm{can}\:\mathrm{give}\:\mathrm{marbles}\:\mathrm{so}\:\mathrm{that}\:{A}\:=\:\mathrm{4} \\ $$$$\mathrm{and}\:{B}\:=\:\mathrm{8},\:\mathrm{and}\:{B}\:\mathrm{can}\:\mathrm{give}\:\mathrm{marbles} \\ $$$$\mathrm{so}\:\mathrm{that}\:{A}\:=\:\mathrm{9}\:\mathrm{and}\:{B}\:=\:\mathrm{3}. \\ $$
Commented by Tinkutara last updated on 12/Aug/17
Thank you very much Sir!
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$

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