What is the decimal equivalent of the 25 digits of hexadecimal number (100....001)16?

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223 + 1

224 + 1

292 + 1

296 + 1

answer is D.

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Detailed Solution

We have to write the decimal equivalent of (100....001)16 .
We can observe that there are 25 digits in 100…..001 in which there are twenty – three 0’s and two 1’s.
Here 1’s are present at the left most and right most positions while 0’s are in between these 1’s.
Suppose we have a hexadecimal number of the form, (abcd)16 , where ‘a’, ‘b’, ‘c’ and ‘d’ are four digits. We have to convert this hexadecimal number into a decimal equivalent. What we do is,
⇒ (abcd)16 = d × 160 + c × 161 + b × 162 + a × 163
Here, 16 is converted into 24 . So,
⇒ (abcd)16 = d × 20 + c × 24 × 1 + b × 24 × 2 + a × 24 × 3
Now, similarly when we will assign 0, 1, 2, …… to (100….001) and write (100....001)16 into its decimal equivalent, we get,
⇒ (100...0001)16 = 1 × 160 + 0 × 161 + 0 × 162 + ....... + 0 × 1622 + 0 × 1623 + 1 × 1624 Converting 16 into 24, we have,
⇒ (100...0001)16 = 1 × 20 + 0 × 24 × 1 + 0 × 24 × 2 + ....... + 0 × 24 × 22 + 0 × 24 × 23 + 1 × 24 × 24
⇒ (100...0001)16 = 1 + 296
⇒ (100...0001)16 = 296 + 1
So, the correct answer is “Option 4”.

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