GATE IT 2004 | Question: 43

Question

The number $(123456)_8$ is equivalent to

• A. $\text{(A72E)}_{16}$ and $(22130232)_4$
• B. $\text{(A72E)}_{16}$ and $(22131122)_4$
• C. $\text{(A73E)}_{16}$ and $(22130232)_4$
• D. $\text{(A62E)}_{16}$ and $(22120232)_4$

Comments

Base 2 = log2 = 1, each bit in the binary representation is considered.
Base 4 = log4 = 2, each unit of  2 bits in the binary representation is considered.
Base 8 = log8 = 3, each unit of 3 bits in the binary representation is considered
Base 16 = log16 = 4, each unit of 4 bits in the binary representation is considered

Answer is (A).

Answer 1

$(123456)_{8} = (001\; 010\;011\; 100\; 101\; 110)_{2} = (00\; \underbrace{1010}_{A}\; \underbrace{0111}_7\; \underbrace{0010}_2\; \underbrace{1110}_E)_{2} = \text{(A72E)}_{16}$
$= (00 \;10\; 10\; 01\; 11\; 00\; 10\; 11\; 10)_{2} = (22130232)_{4}$

So, option (A).

Answer 2

as 8 =2$^3$ so 3 bits of binary are used to represent one unit of octal

so break each unit of (123456)$_8$ into its corresponding 3 bit binary number 

=  (001 010 011 100 101 110)$_2$

as 16=2$^4$ so 4 bits of binary are used to represent one unit in Hexadecimal 

club 4 bits of binary representation of given number from LSB and write corresponding Hexadecimal number.

=(00 1010 0111 0010 1110)$_2$ = (A72E)$_1$$_6$

 

as 4=2$^2$ so 2 bits of binary are used to represent one unit in 4 base number system

club 2 bits of binary representation of given number from LSB and write corresponding Hexadecimal number.

=(00 10 10 01 11 00 10 11 10)$_2$ = (22130232)$_4$

option (A)