Question

Given below are two statements
Statement I: $(543)_6$ is equivalent to $(317)_8$ 
Statement II: The last 4 bits in the binary representation of a multiple of 16 is 1000. 
In light of the above statements, choose the correct answer from the options given below

• A. Both Statement I and Statement II are true
• B. Both Statement I and Statement II are false
• C. Statement I is true but Statement II is false
• D. Statement I is false but Statement II is true

Answer 1

The Correct Option is C

Let's analyze the given statements one by one to determine their validity.

1. Statement I: \( (543)_6 \) is equivalent to \( (317)_8 \).

To verify Statement I, we need to convert each number to a common base, preferably base 10 (decimal), and check if they are equal.

Convert \( (543)_6 \) to decimal:

• Calculate each digit: \( 5 \times 6^2 + 4 \times 6^1 + 3 \times 6^0 \)
• =(5 × 36) + (4 × 6) + (3 × 1)
• = 180 + 24 + 3 = 207

Now, convert \( (317)_8 \) to decimal:

• Calculate each digit: \( 3 \times 8^2 + 1 \times 8^1 + 7 \times 8^0 \)
• =(3 × 64) + (1 × 8) + (7 × 1)
• = 192 + 8 + 7 = 207

Both numbers convert to 207 in decimal, so Statement I is true.

2. Statement II: The last 4 bits in the binary representation of a multiple of 16 is 1000.

A number that is a multiple of 16 in binary ends with four zeros: 0000. Let's examine why:

In binary, multiplying by 16 means shifting left by 4 bits. Thus, any multiple of 16 will have the last four bits as 0000. Therefore, the statement indicating the last 4 bits should be 1000 is false.

Therefore, Statement II is false.

From the analysis above, the correct answer is: Statement I is true but Statement II is false.