Question

What does the hexanumber $E78$ in radix $7$?
A. $12455$
B. $14153$
C. $14356$
D. $13541$

Answer 1

Hint: Here, we have $E78$ in hexadecimal and we have to find the value in radix $7$. Hexadecimal number system is the system which has a base as $16$ (hexa $ = 6$ and deci $ = 10$). In this number system, there are $16$ digits which are used in representing the number in hexadecimal form. For converting the hexadecimal number into radix $7$ we first convert the number in decimal form and then we divide the number by $7$ and note remainder and quotient.

Complete step by step answer:
In mathematics a system of writing or denoting numbers is called a number system and there are two types of number systems which are positional and non- positional number systems. Positional number system is the system in which the value of a digit depends upon its position in the number whereas in a non-positional number system the value of a digit does not vary with its position in the number. Hexadecimal number system is a positional number system.

In this system there are $16$ digits which are used in representing the number in hexadecimal form and it is similar to the decimal number system as the first $10$ digits remain the same in both the number system. But in the hexadecimal number system $10$ is represented as $A$, $11$ as $B$, $12$ as $C$, $13$ as $D$, $14$ as $E$ and $15$ as $F$.So, all the digits in the decimal number are $1,2,3,4,5,6,7,8,9A,B,C,D,E,F$.

Now, we will convert the hexadecimal number $E78$ in decimal form. We convert the hexadecimal number into its decimal equivalent by multiplying each digit with its positional values of $16$. So,
$ \Rightarrow (E78) = 14 \times {16^2} + 7 \times {16^1} + 8 \times {16^0}$
Solving the power of $16$. We get,
$ \Rightarrow (E78) = 14 \times 256 + 7 \times 16 + 8 \times 1$
On multiplying. We get,
$ \Rightarrow (E78) = 3584 + 112 + 8$
Adding the numbers we get,
$ \Rightarrow (E78) = 3704$

Now, we will convert the decimal to radix $7$. For converting decimal into radix, we need to continually divide the number by its radix and with each division write down the remainder and then read from top to bottom the remainder. The top to bottom remainder will be our required result.So, on dividing $3704$ by $7$. We get
$1$ as remainder and $529$ as quotient.
Now $529$ is our dividend and we again divide it by $7$. We get,
$4$ as remainder and $75$ as quotient.
Now $75$ is our dividend and we again divide it by $7$. We get,
$5$as remainder and $10$ as quotient.
Now $10$ is our dividend and we again divide it by $7$. We get,
$3$ as remainder and $1$ as quotient.
Now $1$ is our dividend and we again divide it by $7$. We get,
$1$ as a remainder.
We get our result by reading the remainder from top to bottom i.e, $13541$. Therefore, the equivalent of the hexadecimal number $E78$ in radix $7$ is $13541$.

Hence, option (D) is the correct answer.

Note: In a positional number system the value of any digit depends on the digit whose value is to be determined, position of the digit in the number and base or radix of the number system. Note that a hexadecimal number system has two parts which is an integer and the fraction part. Integer part includes the number to the left of the decimal point and the fraction part indicates the digit to the right of the decimal point.