The base-2 representation of a number is . What is the seventh digit from the left of the representation of in base 8?
Pick one
Solution
The answer is (D). Let the base-2 expansion of a generic integer be
then, if we group the terms three by three starting from the right, collecting a suitable power of 8, we obtain
Since the terms in parentheses are always between 0 and 7, what we obtained is the expression in base 8 of the number . Observing how the digits are grouped, it is easy to conclude that, denoting by the digits of the expansion in base 8 of the number, we have
For the number under examination, operatively, it suffices to group the digits in groups of three starting from the last one on the right, and read (in base 2) the seventh group of digits starting from the left (note that the first group is made up of the single digit 1):
The seventh digit is thus .
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