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We all know even numbers are the ones that end in even digits. How do we analyze even numbers in base 2?

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Note: Even numbers are of the form $2k$ for $k$ an integer. How does multiplying by 2 work in base 2? (Note well that 2 is $(10)_2$ in base 2) –  bonsoon Feb 24 '13 at 8:33

1 Answer 1

up vote 4 down vote accepted

HINT: Look at the last (least significant) digit.

Remember, if $n=(b_mb_{m-1}\dots b_1b_0)_{\text{two}}$, so that the binary digits from left to right are $b_m,\dots,b_0$, then

$$n=\sum_{k=0}^mb_k2^k=2^mb_m+2^{m-1}b_{m-1}+\ldots+2b_1+b_0\;.$$

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Thank you for your answer. So if it ends in $0$, it is even and if it ends in $1$ it is odd, right? –  Gigili Feb 24 '13 at 8:39
    
@Gigili: You’ve got it. –  Brian M. Scott Feb 24 '13 at 8:39

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