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How many different rational numbers between 1/1000 and 1000 can be written either as a power of 2 or as a power of 3, where the exponent is a (possibly negative) integer?

I got 31; is this correct?

Oct 6, 2018

#1
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$$2^9=512$$

$$3^6=729$$

Those are the largest positive powers of 2 and 3 under 1000.

$$2^{-9}=0.001953.....$$

$$3^{-6}=0.0013717.....$$

These are the smallest negative powers greater than  $$\frac{1}{1000}$$

Therefore, we have $$9-(-9)+1=19$$ different powers of 2 between $$\frac{1}{1000}$$ and 1000.

We also have $$6-(-6)+1=13$$ different powers of 3 between $$\frac{1}{1000}$$ and 1000.

This gives us $$19+13=32$$ total integers...

32 is the answer I got but someone is going to have to check over my work :b

Oct 6, 2018
#2
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The problem is symmetric in x vs. 1/x so we can find the number of integers between 0 and 1000 that are either a power of two or 3 and just double the answer.

Note that any non-zero power of 2 is even and any non-zero power of 3 is odd so these two sets are disjoint.

There are 10 powers of 2, 0-9, and 6 non-zero powers of 3 that are less than 100.

We use non-zero power for 3 since we only want to count 1 = 20 = 30 once

That gets us 16 from 0 to 1000.  Doubling this we get 32 but we don't want to count $$1 = \dfrac{1}{1}$$

twice so we subtract 1 from this getting 31 as you found.

Oct 6, 2018
edited by Rom  Oct 6, 2018
edited by Rom  Oct 6, 2018
edited by Rom  Oct 6, 2018
#3
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That is an interesting question and good answers from both guest and Rom.

Thanks :)

Oct 6, 2018