Question

Find the base r if (159)10 =(186) r

Answer 1

Explanation:

(211)x is 2x^2+ 1*x+1 in Decimal System. Therefore either x = 7 or x = -15/2. Obviously x = -15/2 isnot the answer. So, x = 7 is the answer.

Answer 2

Concept:

The number system is a method of representing numerical data.  The conversion of a number from one base to another is known as number system conversion. Binary numbers, decimal numbers, hexadecimal numbers, octal numbers, and other number systems are examples.

Given:

The equation $(159)_{10} = (186)_r$.

Find:

The value of r.

Solution:

$1*10^2+5*10^1+9*10^0 = 1*r^2+8*r^1+6*r^0\\159 = r^2+8r+6\\r^2+8r = 153\\r^2+8r-153=0\\r^2+17r-9r-153=0\\r(r+17)-9(r+17)=0\\(r-9)(r+17)=0\\r=9 \quad or \quad -17$

The base cannot be negative.

Hence, the value of $r$ is $9$.