GATE CS 2024 Set 2 Q.49: Number Representations and Computer Arithmetic

Which of the following is/are EQUAL to 224 in radix-5 (i.e., base-5) notation?

64 in radix-10

100 in radix-8

50 in radix-16

121 in radix-7

Official answer and solution

Solution

Convert $(224)_5$ to base 10 ( $64_{10}$ ) and compare with each option converted to base 10.

First convert $(224)_5$ to decimal (radix-10):

(224)_5 = 2 \times 5^2 + 2 \times 5^1 + 4 \times 5^0 = 50 + 10 + 4 = 64_{10}

Now check each option:

(A) $(64)_{10} = 64_{10}$ (EQUAL).
(B) $(100)_8 = 1 \times 8^2 + 0 \times 8^1 + 0 \times 8^0 = 64_{10}$ (EQUAL).
(C) $(50)_{16} = 5 \times 16^1 + 0 = 80_{10} \neq 64_{10}$ (Not equal).
(D) $(121)_7 = 1 \times 7^2 + 2 \times 7^1 + 1 \times 7^0 = 49 + 14 + 1 = 64_{10}$ (EQUAL).

Final answer

Options (A), (B), (D)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

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