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© 1999-2010 by Francis Edward Su

From the Fun Fact files, here is a Fun Fact at the Easy level:

# Powers of Ten in Base Two and Five

Make two lists; one consists of all powers of 10 in base 2, and the other all powers of 10 in base 5. (Powers of 10 are 10, 100, 1000, etc.)

Fun Fact: for any integer N > 1, there is exactly one number in exactly one of the lists that is exactly N digits long!

Presentation Suggestions:
Show students the first terms in each list: they are
(base 2) 1010, 1100100, 1111101000, ...
(base 5) 20, 400, 13000, 310000, ...

The Math Behind the Fact:
This fact is quite startling when you first see it. Proving it makes a great exercise for a Putnam problem-solving seminar...

Su, Francis E., et al. "Powers of Ten in Base Two and Five." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References:

Subjects:    number theory, other
Level:    Easy
Fun Fact suggested by:   Ravi Vakil
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