Today’s sample problem comes from our friends at ManhattanGMAT. To help prepare for the exam, see if you can solve the problem first, then read on for the correct answer and explanation.
Problem
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) 17/300
(B) 1/15
(C) 1/12
(D) 1/10
(E) 3/25
Solution
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. 162 = 256 and 172 = 289, but 182 = 324. Thus, we have 12 through 172, for 17 perfect squares.
Now, let’s count the cubes. Leave out 13, since we’ve already counted 1 as 12. 53 = 125 and 63 = 216, but 73 = 343. Thus, we have 5 more integer powers (23 through 63, inclusive), for a cumulative total of 22.
What about the fourth powers – 24, 34, etc.? We’ve already counted any of these that matter, because they are also perfect squares: 24 = 42, 34 = 92, etc. We can leave out any higher even powers for the same reason (26 = 82, 28 = 162, etc.).
However, we must consider additional odd powers, continuing to leave out 1 raised to any power. 25 = 32, 35 = 243, but 45 = something greater than 300, as we can see by considering that 45 = 44 × 4 = 162 × 4 = 256 × 4. So we have two more integer powers (25 and 35), for a cumulative total of 24.
Seventh powers: 27 = 128, but 37 = something much greater than 300 (since 37 = 35 × 32 = 243 × 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 25.
Ninth powers: 29 = 128 × 4 = greater than 300. So we can stop here.
Finally, we compute 25/300 = 1/12.
The correct answer is (C).
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