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Topic: Power of 2 divisible by 3
Replies: 19   Last Post: Feb 1, 2006 10:37 PM

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Posts: 5,039
Registered: 12/6/04
Re: Power of 2 divisible by 3
Posted: Jan 30, 2006 2:34 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply wrote:
> I cannot seem to find a power of 2 divisible by 3. I did all of the
> powers up to 4096 in my head. Then I wrote a simple computer program
> to try to find a power of 2 divisible by 3, but it couldn't find any.
> So the number must be above 2^32. But what is the smallest power of 2
> divisible by 3? Does a number x such that log2(x) is an integer and
> x/3 is an integer even exist? Is there a way I might go about proving
> or disproving the existence of such a number?

As mentioned by others, there is no such number.

However, powers of 2 alternate between being congruent to
1 (mod 3) and 2 (mod 3) depending on whether the power is even
or odd.

So what you can say is:

If n is even, all 2**n - 1 are divisible by 3.
If n is odd, all 2**n + 1 are divisible by 3.

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