Question

The base (or radix) of the number system such that the following equation holds is____________.

\(\frac{{312}}{{20}} = 13.1\)

Answer 1

Let the base of system be b.

\(\frac{{{{\left( {312} \right)}_b}}}{{{{\left( {20} \right)}_b}}} = {\left( {13.1} \right)_b}\)

\(\frac{{3 \times {b^2} + 1 \times {b^1} + 2 \times {b^0}}}{{2 \times b + 0 \times {b^0}}} = {\left( {1 \times {b^1} + 3 \times {b^0} + 1 \times {b^{ - 1}}} \right)_{}}\)

\(\frac{{3{b^2} + b + 2}}{{2b}} = b + 3 + \frac{1}{b}\)

\(\frac{{3{b^2} + b + 2}}{{2b}} = \frac{{{b^2} + 3b + 1}}{b}\)

\(3{b^2} + b + 2 = 2{b^2} + 6b + 2\)

b² – 5b = 0

b(b – 5) = 0

b = 0 OR b = 5

\(From\;\frac{{{{\left( {312} \right)}_b}}}{{{{\left( {20} \right)}_b}}} = {\left( {13.1} \right)_b},\;b > 3\)

Therefore, the base (or radix) of the number system such that the following equation holds is 5.