Which of the following options should be the least value of n that sat

Question

Which of the following options should be the least value of n that satisfies the inequality,  2^n > 10^{15}  ?

A. 30
B. 45
C. 60 
D. 75 
E. 90

KarishmaB · Accepted Answer

Whenever you compare terms with exponents, you need to bring either the base or the power equal.

For example, it is very hard to compare

2^{15}  with  3^{12}  (exponent of one term is greater while the power of another is greater) but it is far easier to compare

2^{15}  with  2^{12}

or  2^{15}  with  3^{15}

or 2^{10} with 3^{15} (both exponent and power of one term are smaller so obviously that term is smaller)

So we need to have either the same base or the same power. We need the last value so let's start comparing from option (A)

(A) 30

2^{30} = (2^3)^{10} = 8^{10}

Both exponent and base of  8^{10}  are smaller than the exponent and base of  10^{15} . So n cannot be 30.

(B) 45

2^{45} = (2^3)^{15} = 8^{15}

The base of  8^{15}  is smaller so n cannot be 45.

(C) 60 
Let's try to make exponents equal:

10^{15} = 1000^5

2^{60} = (2^{12})^5 = 1024^5

The base 1024 is greater so  2^{60}  is greater than  10^{15} .

Answer (C)

mikemcgarry · Answer

Dear Sujan Sareen, I'm happy to respond. :-)

This is Mike McGarry, from Magoosh. I gather that you are relatively new here at GMAT Club. I want to give you some advice. This particular forum, the "Ask GMAT Expert" forum, is for general questions (retakes, scheduling, study plans, etc.) For individual content questions, such as an individual math question, you should NOT post it here. You should post this question in the Problem Solving forum: gmat-problem-solving-ps-140/ I would recommend deleting the other version of this question that you are posted: it's considered in particular bad form to post the same question multiple times in different places. Since I am already responding, I am happy to address this question. I will say, when you post a math question, in one of the math forums, it's a good idea to cite the source. For this particular question, it is not at all clear to me that this is GMAT-appropriate question. This is not an approximation that you need to know for the GMAT, but 2^10 is slightly larger than a thousand (10^3). 2^10 = 1024 This is a computer science thing. You see, when a computer talks about kilobytes (kb), that's actually not 1000 bytes, but 1024 bytes. Similarly, a megabyte (Mb) is (1024)^2 bytes, and a gigabyte (Gb) is (1024)^3. Despite the quasi-metric prefixes, everything depends on powers of 2, not powers of 10. So, to make sense of this question, you have to know that 2^10 > 10^3 Now, we simply raise both sides of that inequality to the fifth power. (2^10)^5 > (10^3)^5 2^50 > 10^15 Using this naive approximation, it appears the best answer for n would be 50. In fact, a calculator tells me that 2^50 = 1.1258999 x 10^15 Of the five answer given, n = 45 is too small, so we have to go with n = 60. Again, this could appear as a math problem in other contexts, but I can guarantee that the GMAT does not expect you to know this stuff. Does all this make sense? Mike :-)

OptimusPrepJanielle · Answer

Hi Sujan Sareen,

This is an approximation question, so we would follow a sequence of steps to arrive at the correct approximation.

10^15 = 2^15* 5^15
So, we know that we need to have at least 15 2's in 2^n. - (i)

Next step would be to approximate 5^15.

We can write 5^15 = ((5)^3)^5 = 125^5
We are considering 125, because 125 is closest to 128, which is a power of 2

The closest to 125^5 in terms of powers of 2 would be 128^5
This can be written as (2^7)^5 = 2^35 
Hence we should have 35 2's in 2^n - (ii)

Now adding the 2's in (i) and (ii)
We get 50.

The closest option to this is 60
Hence Option C is the correct answer.

Does this help?

GyMrAT · Answer

This is how i approached the question.

2^n > 10^{15}

can be written as,  2^n > {(2*5)}^{15}

so, we have  2^{(n-15)} > 5^{15}

Now, we know that,  2^5 > 5^2

&  2^{10} > 5^4 , increasing the powers on both sides by 2 times.

Similarly, if we increase the powers on both sides by 8 times, we get

2^{40} > 5^{16}

2^{40} > 5^{15}*5

Now,  2^{(n-15)} > 5^{15}

Hence for LHS > RHS, we need atleast approx  2^{40}*2^3

Therefore  n-15 = 43 , gives  n = 58 , closest answer choice is 60.

Thanks,
GyM

aghosh54 · Answer

2^n>10^{15} 
 2^n>2^{15}*5^{15} 
 2^{n-15}>(2.24)^{30}   (2.24)^2 ]

From here we can say that least value n can take is n-15>30 or n>45.
 Option C is correct

ScottTargetTestPrep · Answer

Recall that 2^10 = 1,024 which is slightly greater than 10^3; therefore,

2^10 > 10^3

Raising both sides of the equation to the fifth power, we have:

(2^10)^5 > (10^3)^5

2^50 > 10^15

Compare this inequality to the inequality given in the question: 2^n > 10^15. Now let’s consider what possible values n could take on to make the inequality true.

If n is at least 50, then 2^n > 10^15. Among the choices, the least value of n such that 2^n > 10^15 is n = 60.

Answer: C

haianh · Answer

(2^3)^{15} = 8^{15} < 10^{15} so 2^{45} < 10^{15} . (2^4)^{15} = 16^{15} > 10^{15} so 2^{60} > 10^{15} . 45 < least value <60. we are given 60, which surely satisfies the condition. Answer: C

DylanD · Answer

1) Factor 10 down to (2^1)*(5^1) and then distribute the exponent of 15
2^n > (2^15)*(5^15)

2) Rephrase to reduce both sides by a factor of (2^15): n = x + 15
(2^x)*(2^15) > (2^15)*(5^15)
(2^x) > (5^15)

3) Consider the implicit ratio and translate into English
2 is two-fifths of 5
2*2 is four-fifths of 5
2*2*2 is eight-fifths of 5
Therefore, for every (5^1), we need (2^3) for the inequality to be true
x = 15*3 = 45 > 15

4) Convert back in terms of n
n = x + 15 > 15
n = 60

ANSWER: C

Which of the following options should be the least value of n that sat

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GMAT Problem Solving (PS) Questions

Which of the following options should be the least value of n that sat

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