Is x 10^10 ? (1) x 2^34 (2) x = 2^35

Question

Is x > 10^10 ?

(1) x > 2^34 
(2) x = 2^35

Bunuel · Accepted Answer

Is x > 10^10 ?

(1) x > 2^34 --> we should compare  2^{34}  and  10^{10}  --> take the square root from both: we should compare  2^{17}  and  10^5=100,000 . Now,  2^{17}=2^{10}*2^7=1,024*128>100,000 . Sufficient.

OR:  2^{34}=(2^{10})^{3.4}=(1,024)^{3.4}>(10^3)^{3.4}=10^{(3*3.4)}=10^{10.2}>10^{10} .

(2) x = 2^35. Since we have the exact numerical value of x we should be able compare to it to 10^10 and answer the question. It really doesn't matter whether 2^35>10^10, the main point is that  we have sufficient information to get the answer . Sufficient.

Answer: D.

EvaJager · Answer

(1) Let's see if  2^{34}>10^{10} .

2^{34}>2^{10}*5^{10} . Divide through by  2^{10} , and we get  2^{24}>5^{10} . Take the square root of both sides:
 2^{12}>5^5 . This we can compute quite easily, and we find that 1024*4=4096 > 625*5= 3125, TRUE.
Sufficient.

(2) We have already seen that (1) is sufficient, obviously (2) is also sufficient.
Just to play with powers, we can check that  2^{35}>10^{10} :
Start again with  2^{35}>2^{10}*5^{10} , divide through by  2^{10} , then  2^{25}>5^{10} . Now we can take the 5th order root of both sides and obtain  2^5>5^2  or 32 > 25, TRUE.

Thus, answer D.

venmic · Answer

Bunuel rocks this is a very nice solution compared to the other ones I ve seen

EvaJager · Answer

Bunuel's logic for testing assumption (2) is excellent. In the present form of the question, one can use that  2^{35}>2^{34} , so once (1) turns out sufficient and necessarily provides the info that  2^{34}>10^{10} , testing (2) is very easy. Maybe, it would have been somehow more challenging to choose a smaller exponent in statement (2), like 33, with which direct comparison would have been not so straightforward, and a time saving approach would need similar logic to Bunuel's.

dabral · Answer

In my personal opinion, I just don't see this question following the style of the GMAT test writers. Unless, whoever wrote this can confirm that they recently saw a similar idea on the exam.

Dabral

reklaw · Answer

Why would you need to compute anything in this problem?

Statement 1 gives us a minimum value of x. It doesn't matter if 10^10 is smaller or larger, it is sufficient to answer the question.
Statement 2 needs no computation either, which Bunuel already pointed out.

Bunuel · Answer

We do need to compare 10^10 and 2^34 in (1). Because if 2^34 were less than 10^10, then the statement wouldn't b sufficient. Consider this:

Is x>10?

(1) x>2. If x=5 then the answer is NO but if x=15, then the answer is YES. Not sufficient.

Hope it's clear.

reklaw · Answer

Crystal.

Classic mistake, which is what is probably REALLY being tested, realised it when re-reading my post.

Thanks for response though.

GMATBeast · Answer

Bunuel,

I solved it a different way- would you mind checking my approach?

I restructured the qstem to is x =,> 2^11 * 5^11?

(1) x > 2^34

- x has 2^11 therefore 2^35 - 2^11 = 2^24
- estimated 2^2 to be 5 and divided 24 by 2 and got 5^12
- x = 2^11 * 5^12 -------- SUFFICIENT

(2) X = 2^35

- SUFFICIENT

Is this approach correct?

mau5 · Answer

A very coarse method but I would do this problem by log.

F.S.1 gives logx > 34 log2 = 34*0.3 = 10.2(approx). As problem statement is asking about whether x>10^10, it boils down to logx>10. Thus suficient.

F.S.2 anyways gives the value of logx = 35*0.3. Again logx>10.
Sufficient.

D.

Not the best method but a pretty quick one. I don't think it is a 700+ level question.

greatps24 · Answer

Good one. Can you please share some source/ tutorial of this method?

mau5 · Answer

No tutorial as such. Just that they gave a power of 10 in this question and also 2^something. So it just struck me. Worked in this question, might not work always!

Bunuel · Answer

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

To find DS questions by Kudos, sort by Kudos here: gmat-data-sufficiency-ds-141/
To find PS questions by Kudos, sort by Kudos here: gmat-problem-solving-ps-140/

blueseas · Answer

hi,

statement 2 is clear we are able to calculate...HENCE SUFFICIENT
for statement 1
2^34=2^10*8^8===>(1
10^10=2^10*5^10===>(2

actually we have to compare 2^34..and 10^10
both of them have 2^10 common...so we have to compare actually 8^8 and 5^10...

5^10=(8-3)^8*25=25*(8-3)^8
lets divide 5^10..with 8^8..==>25*((8-3)/8)^8==>clearly we can see that 25 is multiplied to a very small no.(as bracket number is less than 1,and it has been raised to power 8)==>end resul will be less than 1...==>this proves 8^8 is greater than 5^10....hence...2^34>10^10==>sufficient.

both statements are sufficient..hence D

vad3tha · Answer

(2) is 1 number, so it is sufficient 
(1) compare 2^34 > 10^10 ? 
-> 2^10 * 2^24 > 2^10 * 5^10 
-> 2^24 > 5^10 
-> 2^12 > 5^5 ( square root) 
-> 2^10 * 2^2 > 5^3 * 5^2
-> 1028*4 > 125*25
-> 4112 > 3125 ( correct)

answer D

haihai89 · Answer

Oh  Bunuel , you are always awesome. For me, your approach is always faster than Manhanttan's. 
Highly recommend GMAT club for all GMAT learner!!!

thefibonacci · Answer

whenever there is a 10 (or its multiples) involved, i use log.

is x > 10^10
or log x > 10 log 10 
or log x > 10 ?

1) x > 2^34
log x > 34*log 2
log x > 10.23 (sufficient)

2) has to be sufficient

MathRevolution · Answer

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Is x > 10^10 ?

(1) x > 2^34
(2) x = 2^35

In case of inequality questions, it is important to note that the conditions are sufficient if the range of the question includes the range of the conditions.
There is 1 variable (x) in the original condition. In order to match the number of variables and the number of equations, we need 1 equation. Since the condition 1) and 2) each has 1 equation, there is high chance D is going to be the answer.

In the case of the condition 1), if we do the calculation, we get 10^10<10*(10^3)^3=10*(1000)^3<16*(1024)^3=(2^4)(2^10)^3=2^34. Since the range of the question includes the range of the conditions, the condition 1) is sufficient.

In the case of the condition 2), if x=2^35, the answer can always be either ‘yes’ or ‘no’. Therefore, the condition is sufficient, and the correct answer is D.

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously, there may be cases where the answer is A, B, C or E.

jodoubt · Answer

My working is just a tad different.

Statement 2, enough 
Statement 1,
 x > 10^10
x > 2^10.5^10
 
So x > 2^34
x > 2^10.2^24

Now, my thinking was 5 is 2.5 times more than 2, so 5^10 will be closer to 2^20 but w/o working i cant say higher or lower , whereas 2^21 > 5^10 (definite as its 1 more than double), we have 2^24. So sufficient.
I know its not perfect but saved time for me, of course Bunuel's method is flawless !!.

Is x > 10^10 ? (1) x > 2^34 (2) x = 2^35

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