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8_murik_8 [283]
3 years ago
11

How many different numbers between $\dfrac{1}{1000}$ and $1000$ can be written either as a power of $2$ or as a power of $3$, wh

ere the exponent is an integer?
Mathematics
2 answers:
Pani-rosa [81]3 years ago
7 0

Answer:

31

Step-by-step explanation:

Sever21 [200]3 years ago
3 0

Answer:

31

Step-by-step explanation:

Solution:

First we count the powers of $2$. We can compute $2^9=512$ and $2^{10}=1024$, so the largest power of $2$ which is smaller than $1000$ is $2^9$. Remembering that $2^{-n}$ is defined as $\dfrac{1}{2^n}$, we can see that the smallest power of $2$ which is larger than $\dfrac{1}{1000}$ is $2^{-9}$. Thus, we have the following list of powers of $2$:

\[2^{-9}, 2^{-8}, 2^{-7}, 2^{-6}, 2^{-5}, 2^{-4}, 2^{-3}, 2^{-2}, 2^{-1}, 2^0, \]\[2^1, 2^2, 2^3, 2^4, 2^5, 2^6, 2^7, 2^8, 2^9.\]

There are $19$ numbers in this list ($9$ with negative exponents, $9$ with positive exponents, and one more -- $2^0$).

We can count powers of $3$ in a similar way. The largest power of $3$ smaller than $1000$ is $3^6 = 729$. The smallest power of $3$ larger than $\dfrac{1}{1000}$ is therefore $3^{-6}$, or $\dfrac{1}{729}$. So we have these powers of $3$:$$3^{-6}, 3^{-5}, 3^{-4}, 3^{-3}, 3^{-2}, 3^{-1}, 3^0, 3^1, 3^2, 3^3, 3^4, 3^5, 3^6.$$We don't want to count $3^0$ again, though, because $3^0$ is $1$, which we already counted as $2^0$! There are $12$ other numbers in our list ($6$ with negative exponents and $6$ with positive exponents).

We don't have to worry about any other duplication between the lists, since $\allowbreak 2^1,2^2,2^3,\allowbreak \ldots,2^8,2^9$ are all even and $\allowbreak 3^1,3^2,3^3,\allowbreak 3^4,3^5,3^6$ are all odd. So, we have $19+12 = \boxed{31}$ different numbers in our two lists.

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sveticcg [70]

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<u>Step-by-step explanation:</u>

Let us consider the total no. of apples be 'a'.

Patriarca took 1/6 of the apples. So the remaining apples is,

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Step 1:

His wife takes (1/5) of the rest of the apples. So after she takes the apples the remaining apples will be,

   (5/6) a - (1/5) (5/6) a

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Step 2 :

Her elder son takes (1/4) of the remaining apples. So,

   (4/6) a - (1/4) (4/6) a

   (4/6) a - (1/6) a = (3/6) a

Step 3:

Middle son takes (1/3) of the remaining apples. So,

   (3/6) a - (1/3) (3/6) a

   (3/6) a - (1/6) a = (2/6) a

Step 4:

Younger son takes (1/2) of the remaining apples. So,

   (2/6) a - (1/2) (2/6) a

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After work she takes the remaining 3 apples. So the remaining apples were 3

So,

   (1/6) a = 3

       a = 3 x 6

      a = 18 apples

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7 0
3 years ago
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Answer:

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Step-by-step explanation:

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