Number of Digits

[dtweah]

If N is the number of digits in the number 2^3000 then

(a) N <=500
(b) 500 < N <= 900
(c) 900 < N <= 1000
(d) 1000 < N <=5000
(e) N > 5000

[lunarpower]

this is a totally un-gmat-like question. don't waste your time.

whatever source you got this from, RUN, don't walk, the other way.

[dtweah]

this is a totally un-gmat-like question. don't waste your time.

whatever source you got this from, RUN, don't walk, the other way.

This is A GMAT Question and I have seen questions like this before.
C is the answer

Anybody who knows 10^n has n+1 digits knows that 10^1000 is greater than 8^1000=2^3000 and must have more digits. IF 10^1000 has 1001 digits, 8^1000 cannot have more than 1001 digits. And by inspectiop you know it will have more than 900 digits since 8 is so close to 10. That is how I reasoned it out and chose C in 15 sec.

The OA goes further than my reasoning in showing that 2^10 =1024 > 10^3. Multiply both exponents by 300 and you find 2^3000 > 10^900, which has 901 digits. So 2^3000=8^1000 must have more than 901 digits.

Trust me, I have seen problems on the GMAT 1 million times more difficult that I could not solve in 10 sec. Please stop telling us what can and cannot be asked on GMAT. I think that smacks of phony arrogance!!

So without putting pen to paper in 10 sec this question can be answered

[Ian Stewart]

this is a totally un-gmat-like question. don't waste your time.

whatever source you got this from, RUN, don't walk, the other way.

This is A GMAT Question and I have seen questions like this before.
C is the answer

I've seen most of the published retired GMAT questions, and have never seen this one, so if it is a real GMAT question, I'd be interested to know where it's from. I agree that it can be solved quickly, but the question is not much like any real GMAT question I've encountered, and if something similar were to appear on a real test, it could only show up at the top difficulty level, since not many test takers would answer it correctly, and that's how difficulty level is determined.

[Vemuri]

this is a totally un-gmat-like question. don't waste your time.

whatever source you got this from, RUN, don't walk, the other way.

This is A GMAT Question and I have seen questions like this before.
C is the answer

Anybody who knows 10^n has n+1 digits knows that 10^1000 is greater than 8^1000=2^3000 and must have more digits. IF 10^1000 has 1001 digits, 8^1000 cannot have more than 1001 digits. And by inspectiop you know it will have more than 900 digits since 8 is so close to 10. That is how I reasoned it out and chose C in 15 sec.

The OA goes further than my reasoning in showing that 2^10 =1024 > 10^3. Multiply both exponents by 300 and you find 2^3000 > 10^900, which has 901 digits. So 2^3000=8^1000 must have more than 901 digits.

Trust me, I have seen problems on the GMAT 1 million times more difficult that I could not solve in 10 sec. Please stop telling us what can and cannot be asked on GMAT. I think that smacks of phony arrogance!!

So without putting pen to paper in 10 sec this question can be answered

Chill it buddy I feel your preparation is at the 800 level. Your explanation is very informative. I was literally stuck when I saw this question & reading through your explanation I realised how knowing & implementing certain details (such as 10^n has n+1 digits) can really take us through answering these kind of questions. Keep posting these kind of challenging questions.....relevant to GMAT ofcourse

[dtweah]

this is a totally un-gmat-like question. don't waste your time.

whatever source you got this from, RUN, don't walk, the other way.

This is A GMAT Question and I have seen questions like this before.
C is the answer

I've seen most of the published retired GMAT questions, and have never seen this one, so if it is a real GMAT question, I'd be interested to know where it's from. I agree that it can be solved quickly, but the question is not much like any real GMAT question I've encountered, and if something similar were to appear on a real test, it could only show up at the top difficulty level, since not many test takers would answer it correctly, and that's how difficulty level is determined.

Ian GMAT will ask questions based on arithmetic, exponents geometry etc. 2^3000 is an exponent question. If GMAT could ask that question about h(100)+1 where h is the product of all n numbers for which you and others provided an excellent solution, I am certain they will ask 2^3000, without any doubt.
In terms of difficulty level you be the judge on which of the 2 questions is more difficult.

And speaking of difficulty level, 23 out of 100 people now sitting the GMAT are now scoring more than 48 on Quantity which means GMAT difficulty level can expect to increase. If people are aiming at scoring above 700 which this site hopes to promote, then I think their practise should focus on questions like the h(100)+1 variety I referenced. From postings I have read, many complain that the Q questions are difficult. It may be because they are so used to expect to see only certain question types. The best way to prepare is to train for tough questions and the tricks to handle them. If you don't see them that's fine. If you do, at least you are not surprised. Tricks You, Sura and others provide are valuable. But to argue that 2^3000=8^1000 is beyond GMAT I think does a disservice.

[sureshbala]

If N is the number of digits in the number 2^3000 then

(a) N <=500
(b) 500 < N <= 900
(c) 900 < N <= 1000
(d) 1000 < N <=5000
(e) N > 5000

Folks, this can answered quickly if you have the basics of logarithms and also value of log 2.

log (2^3000)

= 3000(log 2)

= 3000(0.301)

= 903.

Since the log of the given number is 903, the number of digits in it must be 904

Absolute pure mathematics stuff.....

Just thought to share with you guys...

Why don't you guys consider remembering log 2, log 3, log 5 and log 7? They may come in handy....

[Ian Stewart]

This is A GMAT Question and I have seen questions like this before.
C is the answer

If people are aiming at scoring above 700 which this site hopes to promote, then I think their practise should focus on questions like the h(100)+1 variety I referenced. From postings I have read, many complain that the Q questions are difficult. It may be because they are so used to expect to see only certain question types. The best way to prepare is to train for tough questions and the tricks to handle them. If you don't see them that's fine. If you do, at least you are not surprised. Tricks You, Sura and others provide are valuable. But to argue that 2^3000=8^1000 is beyond GMAT I think does a disservice.

I think you may have misinterpreted me. I was saying three things:

-you said that the above was a real GMAT question, and since I've seen almost every published retired GMAT question and didn't recognize it, I was curious about that claim, and I asked where you'd seen it;

-I said that the question was not like other GMAT questions I've seen. That wasn't based on the difficulty level or content of the question; it was based on the style of the question. The structure of the answer choices, for example, almost gives away the answer of C, since only answer choice C offers a narrow range of values. Real GMAT answer choices aren't like that. And if this were a real GMAT question, it's unlikely the power would be as large as 3000 -- it's more likely they'd use a simpler number;

-I did not say the question could not appear on a GMAT; I only said that if it did, it would be a top difficulty level question.

I actually quite like the question, since it can be solved without needing any advanced mathematics (I solved it much as you did, by using estimates based on easy powers), unlike a few questions I've seen recently on this forum that can only be conveniently answered by using knowledge not expected of GMAT test takers. And even if this question might not appear on a real test, there are things a test taker can learn from it that may turn out to be useful.

I do think, however, that it's important for people here to know the likely difficulty level of a question; someone who is at an average level now should be working from questions at the average level that they find challenging, and not from questions that are significantly above their level - until you master the medium level, you won't see any high level questions on an adaptive test. It's for this reason that I pointed out that the question above could only possibly appear at the highest difficulty level, and it's not a question type a test taker should worry about unless they're in the top bracket.

[sureshbala]

Yes, Ian's conclusion makes sense to me....

Since this is an adaptive test, I guess one should worry about questions of this type provided he falls in the top bracket and students who cannot clear avg difficulty level questions may not see question of this standard in the real exam.

[lunarpower]

let me elaborate a bit on what i meant by "not gmat-like" in this context.

this problem is "not gmat-like" in that it requires extremely clever algebraic manipulation, not routine operations.

if the correct answer had been 500 < N < 1000, then this problem could be on the test. as written, NO.

the problem is that the reasoning required to tell whether N is greater or less than 900 is too "mathematician-like". i.e., it is INSIGHT BASED; there is no LOGICAL process whereby one could rationally deduce the idea of using 1024 as a reference number.

the difference is subtle, but EXTREMELY important; this is just not the way the gmat operates. one thing you'll notice is that even the most difficult gmat questions DO NOT require ingenious "insight-style" manipulations. instead, they combine completely routine operations in unexpected or novel ways.

for instance, look at data sufficiency problem #153 in the 11th edition OG. this problem is intimidating, but, in the end, all it requires is extremely basic factoring skills - just factor out the common factor, and you'll realize that the numbers in statement 2 are non-prime.

--

in other words:

your reaction to a genuinely authentic gmat problem should never be "wow, i would never have thought to do that."
if a genuine problem fools you, it should be because you didn't pick up on some signal, or some novel way of presenting a routine operation.

[lunarpower]

The structure of the answer choices, for example, almost gives away the answer of C, since only answer choice C offers a narrow range of values. Real GMAT answer choices aren't like that. And if this were a real GMAT question, it's unlikely the power would be as large as 3000 -- it's more likely they'd use a simpler number;

excellent observations.

ian, just out of curiosity, did you rationally decide that this question wasn't gmat-like, based on these criteria?

i tend to just know whether a question is authentic, malcolm-gladwell-blink-style. i can find these justifications after the fact, but, after seeing enough authentic problems, i can just feel the vibe.
and the vibe just ain't there on this problem.

-I did not say the question could not appear on a GMAT; I only said that if it did, it would be a top difficulty level question.

i say it couldn't appear on the gmat. see above.

here's another analogy:

i could ask you to factor x^4 + (x^2)(y^2) + y^4.
this is do-able; here's what you do.
you "un-simplify" the middle term, into 2(x^2)(y^2) - (x^2)(y^2).
then, you have (x^4 + 2(x^2)(y^2) + y^4) - (x^2)(y^2).
the left-hand trinomial is now a perfect square, so, (x^2 + y^2)^2 - (x^2)(y^2).
this is now a difference of squares, so, (x^2 + y^2 + xy)(x^2 + y^2 - xy).

the point is, though, that the gmat would NEVER ask you to do this, because the "un-simplifying" is just too ingenious and insight-based.

And even if this question might not appear on a real test, there are things a test taker can learn from it that may turn out to be useful.

yes.
concentrate on takeaways at all times.
do not obsess over individual problems.

I do think, however, that it's important for people here to know the likely difficulty level of a question; someone who is at an average level now should be working from questions at the average level that they find challenging, and not from questions that are significantly above their level

all posters should read this comment over and over again until it sinks in.

- until you master the medium level, you won't see any high level questions on an adaptive test.

this isn't quite true. there will be experimental questions that are difficult, and there will also be occasional monkey-wrench (brit: spanner) problems that are more difficult. the test isn't PERFECTLY adaptive; it compromises between perfect adaptiveness and homogeneity across topics, etc., so you may get a couple of random hard ones (and a couple of random easy ones, too) thrown in there.

but the point is that you should worry the most about problems that are AT YOUR LEVEL.
if you're a 600 scorer and you miss a 700+ problem, it doesn't much matter, in the same sense that a young boxer's ranking won't go down appreciably if he loses a bout with the city weight-class champion.

on the other hand, if you're a 600 scorer and you're missing problems at the 600 level, you've got bigger problems.

--

i still maintain that this question, in its current form, could not and would not appear on the test, even at the highest level. "clever insight problems" just don't hack it.

[lunarpower]

and finally, not to be too much of a pedant, but:

why can't anyone figure out how to type "<" ?

you just type "<", but with an underline.

it's much, much better than "<=", which i can't help seeing as a leftward-pointing arrow.

[Ian Stewart]

excellent observations.

ian, just out of curiosity, did you rationally decide that this question wasn't gmat-like, based on these criteria?

i tend to just know whether a question is authentic, malcolm-gladwell-blink-style. i can find these justifications after the fact, but, after seeing enough authentic problems, i can just feel the vibe.
and the vibe just ain't there on this problem.

It was definitely vibe at first; the rationalizations came after the fact. Based on how you describe your reaction to the problem, I'd guess we looked at it in exactly the same way.

i say it couldn't appear on the gmat. see above.

I guess I'd never say never, but I'll agree that I'd be very surprised to see this or a similar problem on a real test. I'm not yet sure I agree that purely 'insight-based' problems never appear on the GMAT, though it's something I'll think about when I look at high-level problems over the next few days, and if I find any examples, I'll post them here. If they do appear, they're certainly rare, but I have occasionally been surprised by the form of some of the most recently retired questions (in GMATFocus, for example).

GMAT question designers do excel at creating questions which offer the opportunity for more than one solution - very often, problems have an 'insight-based' 30-second solution, and a 'brute force' 2-minute solution. Even after working in this field for quite a while, I still find the test interesting - the question design is often ingenious. So 'insight' can certainly help, though it's not required.

- until you master the medium level, you won't see any high level questions on an adaptive test.

this isn't quite true. there will be experimental questions that are difficult, and there will also be occasional monkey-wrench (brit: spanner) problems that are more difficult.

Yes, absolutely agree - I was oversimplifying the situation a bit to make my point quickly.