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What is the least integer n such that

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What is the least integer n such that [#permalink] New post 17 Feb 2017, 02:58
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What is the least integer n such that \(\frac{1}{2^n}\) \(< 0.001\) ?

A) 10

B) 11

C) 500

D) 501

E) There is no such least integer
[Reveal] Spoiler: OA

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Re: What is the least integer n such that [#permalink] New post 21 Feb 2017, 17:01
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Explanation

We rewrite 0.001 as \(\frac{1}{1000}\) .

So if \(\frac{1}{2^n} < \frac{1}{1000}\).

Cross multiplying we can rewrite \(1000 < 2^n\).

So we know \(2^1 = 2, 2^2 =4 ....... 2^1^0 = 1024\).

So if \(n \geq 10\) then the inequality proposed in the question holds.

Hence A is the right answer.
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Re: What is the least integer n such that [#permalink] New post 18 Sep 2017, 15:53
How are we supposed to know that 2^10 = 1024??

Is there a shortcut? Thanks!
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Re: What is the least integer n such that [#permalink] New post 18 Sep 2017, 17:07
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clausen8657 wrote:
How are we supposed to know that 2^10 = 1024??

Is there a shortcut? Thanks!


I remember with Megabyte Gigabyte relation.

1 Gigabyte is \(2^{10}\) Megabyte or 1024
.5 Gigabyte is \(2^9\) Mb or 512

I remember these nubers because these are all multiples of RAM of a computer or cellphone.

Other wise remember \(2^5\) is 32. And \(2^{10}\) is \(32^2\).
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Re: What is the least integer n such that [#permalink] New post 21 Sep 2017, 14:15
how does cross multiplying 2^(1/n) becomes 2^n
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Re: What is the least integer n such that [#permalink] New post 21 Sep 2017, 20:22
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saumya17lc wrote:
how does cross multiplying 2^(1/n) becomes 2^n



it is \(\frac{1}{2^n}\) and not \(2^(1/n)\)
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Re: What is the least integer n such that [#permalink] New post 15 Oct 2017, 15:07
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clausen8657 wrote:
How are we supposed to know that 2^10 = 1024??

Is there a shortcut? Thanks!


It's not a bad idea to memorize powers of 2 up to 2^7
From there, you can keep multiplying by 2 to get bigger powers.

2^7 = 128
2^8 = 256
2^9 = 512
2^10 = 1024



Cheers,
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Re: GRE Math Challenge #80-value of n such that (1/2^n)<0.001 [#permalink] New post 14 Dec 2017, 06:14
sandy wrote:
What is the least integer value of n such that \((1/2^n)<0.001\)?

(A) 10
(B) 11
(C) 500
(D) 501
(E) there is no such least value.



Here
0.001 = 1/1000 now if we can make \(2^n > 1000,\)
then we can write \((1/2^n)<0.001\)

1000 = \(2^3\) * \(5^3\)

\(2^6\) \(<\) \(5^3\) \(<\) \(2^7\)

i.e \(2^3\) * \(2^7\) \(>\) \(2^3\) * \(5^3\)
or \(2^{10} > 1000\)

i.e \(\frac{1}{2^{10}}\) \(<\) 0.001

Hence the minimum value of n = 10
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Re: GRE Math Challenge #80-value of n such that (1/2^n)<0.001 [#permalink] New post 12 May 2018, 07:46
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sandy wrote:
What is the least integer value of n such that \((1/2^n)<0.001\)?

(A) 10
(B) 11
(C) 500
(D) 501
(E) there is no such least value.


We want: 1/(2^n) < 0.001
In other words, we want: 1/(2^n) < 1/1000

Let's start with answer choice A (n = 10)
We get: 1/(2^n) = 1/(2^10)
= 1/1024
Is 1/1024 < 1/1000?
Yes!

Since the question asks us to find the least integer value of n, and since 10 is the smallest answer choice, the correct answer must be A

Cheers,
Brent
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Re: What is the least integer n such that [#permalink] New post 20 Jun 2018, 03:32
how did you cross multiply? please explain
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Re: What is the least integer n such that [#permalink] New post 20 Jun 2018, 04:56
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Ronaksingh wrote:
how did you cross multiply? please explain


Let's do this in steps...

We have the inequality: 1/(2^n) < 1/1000
Since 2^n is always POSITIVE, we can multiply both sides by 2^n to get: 1 < (2^n)/1000
Also, since 1000 is POSITIVE, we can multiply both sides by 1000 to get: 1000 < 2^n

Here's a video on dealing with inequalities like this:

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Re: What is the least integer n such that [#permalink] New post 20 Nov 2018, 15:27
sandy wrote:
Explanation

We rewrite 0.001 as \(\frac{1}{1000}\) .

So if \(\frac{1}{2^n} < \frac{1}{1000}\).

Cross multiplying we can rewrite \(1000 < 2^n\).

So we know \(2^1 = 2, 2^2 =4 ....... 2^1^0 = 1024\).

So if \(n \geq 10\) then the inequality proposed in the question holds.

Hence A is the right answer.

Please explain, is it n>=10 or n>10. How do we sure that n=10, because the question says 1/2^n<1/10^3?
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Re: What is the least integer n such that [#permalink] New post 20 Nov 2018, 15:30
GreenlightTestPrep wrote:
Ronaksingh wrote:
how did you cross multiply? please explain


Let's do this in steps...

We have the inequality: 1/(2^n) < 1/1000
Since 2^n is always POSITIVE, we can multiply both sides by 2^n to get: 1 < (2^n)/1000
Also, since 1000 is POSITIVE, we can multiply both sides by 1000 to get: 1000 < 2^n

Here's a video on dealing with inequalities like this:

So, Is The answer A or B?
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Re: What is the least integer n such that [#permalink] New post 20 Nov 2018, 16:48
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Dear An,

the answer is A. Sandy pointed out that n=10 which is \(2^{10}\) still holds, simply because considering that the stem tells us the least AND that does not exist a number as a result of \(2^n = 1000\), we must have that \(2^{10} = 1024\), we could say that \(n > = 10\)

But these are nuances that you do know and acquire with practice. You develop a sort of instinct.

Be flexible in your approach. You can gain only benefits.

Regards
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Re: What is the least integer n such that [#permalink] New post 21 Nov 2018, 16:52
Carcass wrote:
Dear An,

the answer is A. Sandy pointed out that n=10 which is \(2^{10}\) still holds, simply because considering that the stem tells us the least AND that does not exist a number as a result of \(2^n = 1000\), we must have that \(2^{10} = 1024\), we could say that \(n > = 10\)

But these are nuances that you do know and acquire with practice. You develop a sort of instinct.

Be flexible in your approach. You can gain only benefits.

Regards


I got it now. I was confused with this basic thing.

1024 > 1000
so
1/1024 < 1/1000
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Re: What is the least integer n such that [#permalink] New post 23 Nov 2018, 13:02
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Carcass wrote:


What is the least integer n such that \(\frac{1}{2^n}\) \(< 0.001\) ?

A) 10

B) 11

C) 500

D) 501

E) There is no such least integer


0.001 means 10^-3 means 1/1,000

So anything larger than 1,000

2^10 = 1,024

Answer choice A
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Re: What is the least integer n such that [#permalink] New post 27 Nov 2018, 21:37
it was one of questions in the exam
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Re: What is the least integer n such that [#permalink] New post 03 Dec 2018, 05:07
Alternative solution. I multiplied by 1000 in the beginning so:

1000 / 2^n < 1
8*125 / 2^n < 1 ( factoring the 2s out of 1000 )
=>
2^n > 8*125 ( the denomenator should be > numerator ) then divide all by 8=2^3
2^(n-3) > 125
=>
2^7 > 128
=>
n-3 = 7
=>
n = 10
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Re: What is the least integer n such that [#permalink] New post 18 Aug 2019, 04:00
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1/2^n > 0.001
2^n > 1/0.001
2^n > 1000

now let,
2^7 = 128
2^8 = 256
2^9 = 512
2^10 = 1024 >1000

so ans is A
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Re: What is the least integer n such that [#permalink] New post 19 Aug 2019, 09:06
this is actually hard
Re: What is the least integer n such that   [#permalink] 19 Aug 2019, 09:06
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