Question

In base x system , 1/5 and bar 0.17 are numerals for the same number . Find x. ​

Answer 1

Given :  base x system , 1/5 and baar 0.17 are numerals for the same number

To find :  x

Solution:

baar 0.17  =  $0.\overline{17}$

Let say N = $0.\overline{17}$   = 1/5

base x system

Multiplying by  x²

x²N  = $17.\overline{17}$

=>  N(x² - 1 ) = 17

=> N = 17/(x² - 1 )

=> 17/(x² - 1 ) = 1/5

Converting  17 & 5 into Decimal

17 = 1*x¹  + 7*x⁰   = x + 7

5 = 5*x⁰   = 5

( x  + 7   )/ (x² - 1 )  = 1/5

=> 5x  + 35   = x² - 1

=> x² -  5x   - 36 = 0

=> x²  - 9x + 4x  - 36 = 0

=> x(x - 9) + 4(x - 9) = 0

=> (x + 4) (x - 9) = 0

=> x  = 9

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