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S98-18

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S98-18 [#permalink]

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An integer between 1 and 300, inclusive, is chosen at random. What is the probability that the integer so chosen equals an integer raised to an exponent that is an integer greater than 1?

A. \(\frac{17}{300}\)
B. \(\frac{1}{15}\)
C. \(\frac{2}{25}\)
D. \(\frac{1}{10}\)
E. \(\frac{3}{25}\)
[Reveal] Spoiler: OA

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Official Solution:

An integer between 1 and 300, inclusive, is chosen at random. What is the probability that the integer so chosen equals an integer raised to an exponent that is an integer greater than 1?

A. \(\frac{17}{300}\)
B. \(\frac{1}{15}\)
C. \(\frac{2}{25}\)
D. \(\frac{1}{10}\)
E. \(\frac{3}{25}\)


First, make sure that you grasp the question. To find the desired probability, we need to count the integers between 1 and 300 that fit the given constraint. (We will then divide this count by 300, to determine the final answer.)

The constraint is worded in a confusing way, so we should attempt to reword it. If necessary, put in sample numbers to make the conditions make sense. We need an "integer raised to an exponent that is an integer greater than 1." In other words, we need an integer raised to the 2nd, 3rd, 4th, etc. power. In even other words, we need perfect squares, perfect cubes, etc. that are between 1 and 300, inclusive.

Now we need to count these perfect squares, etc. in an efficient way. Let's start with the squares. What's the biggest perfect square less than 300? Test numbers if necessary. \(16^2 = 256\) and \(17^2 = 289\), but \(18^2 = 324\). Thus, we have 12 through 172, for 17 perfect squares.

Now, let's count the cubes. Leave out \(1^3\), since we've already counted 1 as \(1^2\). \(5^3 = 125\) and \(6^3 = 216\), but \(7^3 = 343\). Thus, we seem to have 5 more integer powers (\(2^3\) through \(6^3\), inclusive) - but careful! \(4^3 = 64\), which we've already counted as a square (\(8^2\)), so we only have 4 more integer powers. This gives us a cumulative total of 21.

What about the fourth powers - \(2^4\), \(3^4\), etc.? We've already counted any of these that matter, because they are also perfect squares: \(2^4 = 42\), \(3^4 = 9^2\), etc. We can leave out any higher even powers for the same reason (\(2^6 = 8^2\), \(2^8 = 16^2\), etc.).

However, we must consider additional odd powers, continuing to leave out 1 raised to any power. \(2^5 = 32\), \(3^5 = 243\), but \(4^5 =\) something greater than 300, as we can see by considering that \(4^5 = 4^4 \times 4 = 16^2 \times 4 = 256 \times 4\). So we have two more integer powers (\(2^5\) and \(3^5\)), for a cumulative total of 23.

Seventh powers: \(2^7 = 128\), but \(3^7 =\) something much greater than 300 (since \(3^7 = 3^5 \times 3^2 = 243 \times 9\)). Be sure to stop calculating when you see that the result is outside the bounds of the problem.

We have 1 more power, for a cumulative total of 24.

Ninth powers: \(2^9 = 128 \times 4 =\) greater than 300. So we can stop here.

Finally, we compute \(\frac{24}{300} = \frac{2}{25}\).


Answer: C
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
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New post 23 Mar 2015, 01:57
Any other method.....it is too long..?

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New post 21 Jul 2015, 08:14
any other simpler/faster method?

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New post 26 Jul 2016, 20:24
is it a real gmat question ?
it takes twoo much time.

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Re: S98-18 [#permalink]

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New post 26 Dec 2016, 08:13
is there any shortcut method? Seems the only method to get this sorted out is working through all options. is it really he case with real GMAT questions?

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New post 26 Dec 2016, 10:48
vnitnagpur wrote:
is there any shortcut method? Seems the only method to get this sorted out is working through all options. is it really he case with real GMAT questions?



Check here: an-integer-between-1-and-300-inclusive-is-chosen-at-random-87439.html
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

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Re: S98-18 [#permalink]

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New post 15 Sep 2017, 02:45
can anyone please explain me this quant ?

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Re: S98-18 [#permalink]

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New post 15 Sep 2017, 02:48

Kudos [?]: 128901 [0], given: 12183

Re: S98-18   [#permalink] 15 Sep 2017, 02:48
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