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Question Number 121996 by Anuragkar last updated on 13/Nov/20

$$\lfloor x\rfloor +\lfloor 2x\rfloor +\lfloor 4x\rfloor +\lfloor 8x\rfloor +\lfloor 16x\rfloor +\lfloor 32x\rfloor =123456$$$$findallvaluesofxforwhichthisrelationholds?$$

Answered by prakash jain last updated on 13/Nov/20

$$x=n+f$$$$n(1+2+4+...+32)$$$$\lfloor 2f\rfloor +\lfloor 4f\rfloor +...+\rfloor 32f\rfloor =123456-63n$$$$=63(1959-n)+39$$$$aslhs<63$$$$\lfloor 2f\rfloor +\lfloor 4f\rfloor +..\lfloor 32f\rfloor =39$$$$f⩽\frac{1}{2}\max \mathrm{lhs}=1+2+4+8+16=31$$$$\lfloor 2f\rfloor =1$$$$\lfloor 4f\rfloor +\lfloor 8f\rfloor +\lfloor 16f\rfloor +\lfloor 32f\rfloor =38$$$$f⩾\frac{3}{4}\mathrm{lhs}>3+6+12+24$$$$\frac{1}{2}<f<\frac{3}{4}$$$$\lfloor 8f\rfloor +\lfloor 16f\rfloor +\lfloor 32f\rfloor =36$$$$f⩽\frac{5}{8}$$$$\frac{5}{8}<f<\frac{3}{4}\Rightarrow >\mathrm{or}\frac{10}{16}<f<\frac{12}{16}$$$$\lfloor 16f\rfloor +\lfloor 32f\rfloor =31$$$$f⩾\frac{11}{16}\Rightarrow 11+22>31$$$$\frac{10}{16}<f<\frac{11}{16}$$$$[32f\rfloor =21$$$$\Rightarrow f=\frac{21+g}{32}0⩽g<1$$$$x=1959+\frac{21+g}{32}(0⩽g<1)$$

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