GATE CS 1996 Q2.21 — Combinational Circuits

To identify the function of the circuit, let us analyze the relation between the 4-bit binary input $b_3b_2b_1b_0$ (values $0$ to $15$) and the 5-bit output $d_4d_3d_2d_1d_0$:

• For a radix-12 (base-12) representation, any value $N \in [0, 15]$ is represented as: $$N = 12 \times d_4 + (d_3d_2d_1d_0)_{2}$$
• The most significant digit $d_4$ (the quotient when divided by 12) is:
  • $d_4 = 0$ for $N < 12$ (inputs $0000_2$ to $1011_2$)
  • $d_4 = 1$ for $N \ge 12$ (inputs $1100_2$ to $1111_2$)
• The condition for $d_4 = 1$ is precisely when $b_3 = 1$ and $b_2 = 1$, which is implemented as $d_4 = b_3 \cdot b_2$ using an AND gate.
• When $d_4 = 1$, the circuit subtracts $12$ ($1100_2$) from the input to obtain the remainder $d_3d_2d_1d_0$.

Therefore, the circuit performs a Binary to radix-12 conversion.