Two-different-two-digit-natural-numbers-are-written-beside-each-other-such-that-the-larger-number-is-written-on-the-left-When-the-absolute-difference-of-the-two-numbers-is-subtracted-from-the-four-di

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Question Number 113005 by Aina Samuel Temidayo last updated on 10/Sep/20

$$\mathrm{Two}\mathrm{different}\mathrm{two}-\mathrm{digit}\mathrm{natural}$$$$\mathrm{numbers}\mathrm{are}\mathrm{written}\mathrm{beside}\mathrm{each}$$$$\mathrm{other}\mathrm{such}\mathrm{that}\mathrm{the}\mathrm{larger}\mathrm{number}$$$$\mathrm{is}\mathrm{written}\mathrm{on}\mathrm{the}\mathrm{left}.\mathrm{When}\mathrm{the}$$$$\mathrm{absolute}\mathrm{difference}\mathrm{of}\mathrm{the}\mathrm{two}$$$$\mathrm{numbers}\mathrm{is}\mathrm{subtracted}\mathrm{from}\mathrm{the}$$$$\mathrm{four}-\mathrm{digit}\mathrm{number}\mathrm{so}\mathrm{formed},\mathrm{the}$$$$\mathrm{number}\mathrm{obtained}\mathrm{is}5481.\mathrm{What}\mathrm{is}\mathrm{the}$$$$\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{two}-\mathrm{digit}\mathrm{numbers}?$$

Answered by floor(10²Eta[1]) last updated on 11/Sep/20

$${\mathrm{n}}_1=10\mathrm{a}+\mathrm{b}$$$${\mathrm{n}}_2=10\mathrm{c}+\mathrm{d}$$$${\mathrm{n}}_1>{\mathrm{n}}_2$$$${\mathrm{n}}_3=\mathrm{abcd}={10}^3\mathrm{a}+{10}^2\mathrm{b}+10\mathrm{c}+\mathrm{d}={100\mathrm{n}}_1+{\mathrm{n}}_2$$$${\mathrm{n}}_3-({\mathrm{n}}_1-{\mathrm{n}}_2)=5481$$$${\mathrm{n}}_1+{\mathrm{n}}_2=?$$$${100\mathrm{n}}_1+{\mathrm{n}}_2-{\mathrm{n}}_1+{\mathrm{n}}_2={99\mathrm{n}}_1+{2\mathrm{n}}_2=5481$$$$\Rightarrow {\mathrm{n}}_1\mathrm{is}\mathrm{odd}\therefore \mathrm{b}\in \{1,3,5,7,9\}$$$$\mathrm{since}{\mathrm{n}}_2⩾10:$$$$5481={99\mathrm{n}}_1+{2\mathrm{n}}_2⩾{99\mathrm{n}}_1+20$$$${\mathrm{n}}_1⩽55$$$${\mathrm{n}}_2⩽98:$$$$5481={99\mathrm{n}}_1+{2\mathrm{n}}_2⩽{99\mathrm{n}}_1+196$$$$53.3⩽{\mathrm{n}}_1\Rightarrow {\mathrm{n}}_1⩾54$$$$\mathrm{but}\mathrm{we}\mathrm{know}\mathrm{that}{\mathrm{n}}_1\mathrm{is}\mathrm{odd}\mathrm{so}\mathrm{the}$$$$\mathrm{only}\mathrm{possible}\mathrm{case}\mathrm{is}{\mathrm{n}}_1=55$$$${\mathrm{n}}_1=55\Rightarrow 5445+{2\mathrm{n}}_2=5481$$$$\Rightarrow {\mathrm{n}}_2=18$$$$\mathrm{so}{\mathrm{n}}_1+{\mathrm{n}}_2=73$$

Commented by Aina Samuel Temidayo last updated on 11/Sep/20

$$\mathrm{Ok}.\mathrm{Thanks}.$$

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