GATE CS 2024 Set 2 Q39 — Number Representations (Fixed and Floating Point)

To find which of the given options are equal to $(224)_5$ (radix-5), we first convert $(224)_5$ to its decimal (base-10) equivalent:

Now, let us evaluate each option by converting them to decimal:

• Option A: $64$ in radix-10
  This is already in base-10: $64_{10}$. Since $64_{10} = 64_{10}$, this option is correct.
• Option B: $100$ in radix-8
  $$(100)_8 = 1 \times 8^2 + 0 \times 8^1 + 0 \times 8^0 = 64_{10}$$ Since $64_{10} = 64_{10}$, this option is correct.
• Option C: $50$ in radix-16
  $$(50)_{16} = 5 \times 16^1 + 0 \times 16^0 = 80_{10}$$ Since $80_{10} \neq 64_{10}$, this option is incorrect.
• Option D: $121$ in radix-7
  $$(121)_7 = 1 \times 7^2 + 2 \times 7^1 + 1 \times 7^0 = 49 + 14 + 1 = 64_{10}$$ Since $64_{10} = 64_{10}$, this option is correct.