> I have no mathematical proof, and I'm not the one to find
> it. 

True. Google is the one to find 
it:http://www.exploringbinary.com/a-pattern-in-powers-of-ten-and-their-binary-equivalents/

"The pattern is easy to explain. A nonnegative power of ten is a multiple of a 
power of five and a power of two: 10n = 5n * 2n. A power of five always ends in 
’5′, so it’s odd — its binary representation always end in ’1′. When you 
multiply by a power of two, you shift the power of five left by n bits, which 
adds n trailing 0s. So the binary representation ends with a ’1′ followed by n 
0s, which looks like the power of ten!"



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