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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]  17 Feb 2017, 02:58
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Question Stats:

77% (00:57) correct 22% (01:23) wrong based on 63 sessions

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]  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]  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]  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]  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]  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]  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,
Brent
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Re: What is the least integer n such that [#permalink]  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]  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]  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]  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]  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]  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]  23 Nov 2018, 13:02
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

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Re: What is the least integer n such that [#permalink]  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]  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
Re: What is the least integer n such that   [#permalink] 03 Dec 2018, 05:07
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