## Number system-2

Sol:

672810 = 1A4816

#### Find the possible basses of the following numbers

a) $\sqrt{45}=5$            b) $\frac{66}{6}=11$

a) $\sqrt{4&space;*&space;b^{1}&space;+&space;1&space;*&space;b^{0}}&space;=&space;5&space;*&space;b^{0}$

$\sqrt{4b+1}&space;=&space;5$

$4b=24$

$b=6$

Result is $\sqrt{41}_{6}&space;=&space;5_{6}$

b)

$\frac{6&space;*&space;b^{1}&space;+6&space;*&space;b^{0}}{6&space;*&space;b^{0}}&space;=&space;1&space;*&space;b^{1}+&space;1&space;*&space;b^{0}$

6b+6 = 6b+6

The above equation satisfies for any value of b mathematically. But as per the number base rule the base value should not be less than the first value i.e. 6 in the above equation.

So the above equation satisfies for any base which is >6 for example 7.

#### In a positioned weighted system if P & Q are the successive bits in the equations PQ=25 and QP=31. find out P and Q?

sol:

First it is indicated that the Q and P are successive digits i.e.

Q=P+1.

Pb + Q = 25      –1

Qb + P  =31    —  2

Substituting the Q value in equation -2 then

(P+1)b +P = 31

Pb+b+P = 31 — 3

By subtracting equation 1 from equation 3 then

b-1 =6

b=7.

by substituting b value in equation 3

8P = 24

P=3

Q=P+1=3+1=4.

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