The representation of the value of a 16-bit unsigned integer $X$ in hexadecimal number system is BCA9. The representation of the value of $X$ in octal number system is

To convert the hexadecimal number $\text{BCA9}_{16}$ to octal:

1. First, convert each hexadecimal digit to its $4$-bit binary equivalent:

   • $\text{B}_{16} = 1011_2$

   • $\text{C}_{16} = 1100_2$

   • $\text{A}_{16} = 1010_2$

   • $9_{16} = 1001_2$

   Combining these gives the 16-bit binary representation:

   $$X = 1011\;1100\;1010\;1001_2$$

2. Group the binary digits into sets of $3$ bits starting from the least significant bit (right to left), padding with leading zeros if necessary:

   $$001 \quad 011 \quad 110 \quad 010 \quad 101 \quad 001$$

3. Convert each 3-bit group to its octal digit:

   • $001_2 = 1$

   • $011_2 = 3$

   • $110_2 = 6$

   • $010_2 = 2$

   • $101_2 = 5$

   • $001_2 = 1$

Thus, the octal representation is 136251, which corresponds to option D.