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Some more important examples: 

1. Find the value of x for the equality

Solution:

The solution is not possible.

As,      in L.H.S, x > 3

And    in R.H.S, x < 3

So, no values of x are possible.

2. The base or radix of the number system such that the following equation holds:

Solution:

First, convert it to decimal.

r = 0 is not possible, (base 0). But r = 5 is possible. So, r = 5

3. Find radix or base of the number system such that

is valid. Find r for which it is valid.

Solution:

So, definitely r ≥ 7 because ${( )}_{\mathrm{r}}$has 0 to r – 1.

So, for r ≥ 7 or r > 6, it is valid.

We can solve it another way by converting it to a decimal system.

And highest digit is 6, so, the value of the base can be any number r ≥ 7or r > 6.

Consider the above equation where x and y are unknown. Find the number of possible solutions i.e. how many possible solutions for x and y.

Important Note:

Radix or base never 0 or fractional or negative number, it is always an integer number.

Solution:

From ${\left(43\right)}_{\mathrm{x}}$ it is clear that x should be greater than 4 because ${( )}_{\mathrm{x}}$ then it has x different number 0 – x – 1.

So, x > 4 or x ≥ 5.

And ${\left(\mathrm{y}3\right)}_8$ from this, it is clear that y should be less than 8 because ${( )}_8$ contain 0 – 7 values.

So, y < 8

Now we convert it to decimal to know the exact number of solution for x and y for which

So,

Answer:  5 possible values of x and y