Given below are two statements:

Statement I: (1011101)₂ = (135)₈

Statement II: (71)₈ = (111001)₂

In the light of the above statements, choose the most appropriate answer from the options given below:

The correct option is Both Statement I and Statement II are true.

Let us verify both statements step-by-step by converting the numbers between binary (base 2) and octal (base 8) systems.

Verification of Statement I:
Statement I claims that ${\left(1011101\right)}^2={\left(135\right)}^8$.

To convert a binary number to octal, we group the binary digits (bits) in groups of three, starting from the rightmost digit (least significant bit). If the leftmost group has fewer than three bits, we pad it with leading zeros.

The binary number is: $1011101$

Grouping into sets of three from right to left:
Group 1 (rightmost): $101$
Group 2 (middle): $011$
Group 3 (leftmost): $1$ (which becomes $001$ when padded with leading zeros)

Now, we convert each 3-bit binary group into its equivalent octal (decimal) digit:
${\left(001\right)}_2=0\times 2^2+0\times 2^1+1\times 2^0=1$
${\left(011\right)}_2=0\times 2^2+1\times 2^1+1\times 2^0=3$
${\left(101\right)}_2=1\times 2^2+0\times 2^1+1\times 2^0=5$

Combining the octal digits in order from left to right gives:
$135$

Thus, ${\left(1011101\right)}_2={\left(135\right)}_8$. Statement I is true.

Verification of Statement II:
Statement II claims that ${\left(71\right)}^8={\left(111001\right)}^2$.

To convert an octal number to binary, we convert each individual octal digit into its corresponding 3-bit binary equivalent:

For the digit 7:
$7_8={\left(111\right)}_2$ (since $4+2+1=7$)

For the digit 1:
$1_8={\left(001\right)}_2$ (represented as a 3-bit group)

Combining the binary groups in order gives:
$111001$

Thus, ${\left(71\right)}_8={\left(111001\right)}_2$. Statement II is true.

Therefore, both Statement I and Statement II are true statements.