Question

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

• A. 64 in radix-10
• B. 100 in radix-8
• C. 50 in radix-16
• D. 121 in radix-7

Hint:

To convert a number from any radix to decimal, multiply each digit by the base raised to its positional power and sum up the results.

Answer 1

The Correct Option is A

The given number \(224\) in radix-5 can be converted to decimal (radix-10) as follows: \[ 224_5 = 2 \cdot 5^2 + 2 \cdot 5^1 + 4 \cdot 5^0 = 50 + 10 + 4 = 64. \]
Now, check each option:
Option (A): \(64\) in radix-10 matches the decimal value of \(224_5\). Hence, (A) is correct.
Option (B): Convert \(100_8\) (radix-8) to decimal: \[ 100_8 = 1 \cdot 8^2 + 0 \cdot 8^1 + 0 \cdot 8^0 = 64. \] This matches \(224_5\). Hence, (B) is correct.
Option (C): Convert \(50_{16}\) (radix-16) to decimal: \[ 50_{16} = 5 \cdot 16^1 + 0 \cdot 16^0 = 80. \] This does not match \(224_5\). Hence, (C) is incorrect.
Option (D): Convert \(121_7\) (radix-7) to decimal: \[ 121_7 = 1 \cdot 7^2 + 2 \cdot 7^1 + 1 \cdot 7^0 = 49 + 14 + 1 = 64. \] This matches \(224_5\). Hence, (D) is correct. Final Answer: \[ \boxed{\text{(A), (B), (D)}} \]