GATE CS 2024 Set 2 — Question 49

To determine which options are equivalent to $(224)_5$, we first convert it to decimal (base-$10$):$$(224)_5 = 2 \times 5^2 + 2 \times 5^1 + 4 \times 5^0 = 2 \times 25 + 2 \times 5 + 4 \times 1 = 50 + 10 + 4 = (64)_{10}$$Now, we evaluate each option by converting it to decimal:

• Option (A): $64$ in radix-$10$ is $(64)_{10}$. This matches.
• Option (B): $100$ in radix-$8$ is $1 \times 8^2 + 0 \times 8^1 + 0 \times 8^0 = 64 + 0 + 0 = (64)_{10}$. This matches.
• Option (C): $50$ in radix-$16$ is $5 \times 16^1 + 0 \times 16^0 = 80 + 0 = (80)_{10}$. This does not match.
• Option (D): $121$ in radix-$7$ is $1 \times 7^2 + 2 \times 7^1 + 1 \times 7^0 = 49 + 14 + 1 = (64)_{10}$. This matches.

Since options (A), (B), and (D) all equal $(64)_{10}$, they are the correct choices.