Answer: A) 63
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Explanation:
Whenever you have a series of nothing but '1's like this, the base 10 equivalent is always going to be 1 less than a power of 2.
The number will be of the form [tex]2^n-1[/tex] where n is a nonnegative integer.
If you're familiar with the list of powers of 2, then you'll likely realize that 64 is a power of 2 since [tex]2^6 = 64[/tex]. One less than this is [tex]2^6-1 = 64-1 = 63[/tex]
This leads to choice A as the answer.
The other answer choices are not one less than a power of 2, so we can cross those off the list.
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Further explanation:
Each slot in the binary number given represents a power of 2. We multiply those '1's by a power of 2 and add them up like so
1*(2^5) + 1*(2^4) + 1*(2^3) + 1*(2^2) + 1*(2^1) + 1*(2^0)
1*32 + 1*16 + 1*8 + 1*4 + 1*2 + 1*1
32 + 16 + 8 + 4 + 2 + 1
63
Summing the first n powers of 2 leads to the result [tex]2^{n}-1[/tex]
So, summing the first 6 powers of 2 leads to [tex]2^n-1 = 2^6-1 = 63[/tex]
