FOR FREE YEAR SOLVED

Some important examples:

For base 4, the maximum range of digits is 3 (0, 1, 2, 3). Since, $4 = 2^2$, represent each digit in base 4 with 2 digits.

In order to convert the numbers from base 4 to base 8, first convert it to binary, then convert it to octal.

Which of the following options represent the correct value of x?

a) 8

b) 10

c) 12

d) > 4

Solution:

So, it holds true for x = 8

As we know if the base is r then the number range of this base between 1 to r - 1. So, here number is 144 so, obviously base will be > 4.   

Since, option (d) represents x > 4. So, we will be considering other options as well.

Say, x = 12

So, correct answer is option (d), i.e. x > 4.

3. What will be the possible values of x and y?

Solution:

We can do one way that every time we convert it to decimal because from decimal it is easy to work and converts.

Factors of 35 are (1, 5, 7, 35)

Now, x = 7 and y = 5 is not possible as, with base 5, digits from 0 to 4 only can be represented.

So, correct values of x and y will be x = 5 and y = 7

4. Find the values of r

Answer:

We can do one way that every time we convert it to decimal because from decimal it is easy to work and converts.

From this equation, we can’t define but both sides equal.

[Both side square]

It is true for all base r ≥ 3.

See the number is 121 so, it is not possible to binary here because ${(\hspace{0.17em})}_{\mathrm{r}}$ has 0 – r – 1 different digit. It must be r ≥ 3 or r > 2

So, for all r ≥ 3 or r > 2 is true.