Problem Solving

j_shreyans wrote: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)2/25
D)1/10
E)3/25

In other words, what it the probability that the selected number can be written as k² (aka square), as k³ (aka cube), k�, , etc.

Squares between 1 and 300 inclusive
1 (= 1²)
4 (= 2²)
9 (= 3²)
.
.
.
225 (= 15²)
256 (= 16²)
289 (= 17²)
324 (= 18²)
TOTAL SQUARES = 17

Cubes between 1 and 300 inclusive
1 (= 1³) [already counted as 1²]
8 (= 2³)
27 (= 3³)
64 (= 4³) [already counted as 8²]
125 (= 5³)
216 (= 6³)
343 (= 7³)
TOTAL (uncounted) CUBES = 4

4th powers between 1 and 300 inclusive
1 (= 1�) [already counted as 1²]
16 (= 2�) [already counted as 4²]
81 (= 3�) [already counted as 9²]
256 (= 4�) [already counted as 16²]
625(= 5�)
TOTAL (uncounted) 4th POWERS = 0

5th powers between 1 and 300 inclusive
1 (= 1�) [already counted as 1²]
32 (= 2�)
243 (= 3�)
1024(= 4�)
TOTAL (uncounted) 5th POWERS = 2

6th powers between 1 and 300 inclusive
1 (= 1�) [already counted as 1²]
64 (= 2�) [already counted as 8²]
729(= 3�)
TOTAL (uncounted) 6th POWERS = 0

7th powers between 1 and 300 inclusive
1 (= 7�) [already counted as 1²]
128 (= 2�)
300+(= 3�)
TOTAL (uncounted) 7th POWERS = 1

8th powers between 1 and 300 inclusive
1 (= 1�) [already counted as 1²]
256 (= 2�) [already counted as 16²]
300+ (= �)
TOTAL (uncounted) 8th POWERS = 0

9th powers between 1 and 300 inclusive
1 (= 7�) [already counted as 1²]
512(= 2�)
TOTAL (uncounted) 9th POWERS = 0

Phew!!!

TOTAL # of squares, cubes, etc = 17+4+0+2+0+0+1+0+0
= 24

So, P(selected number is a square, cube, etc) = 24/300
= [spoiler]2/25[/spoiler]
= C

Cheers,
Brent

Hi j_shreyans,

This question does require a bit of work/knowledge, but there is a Number Property that can save you some time and the prompt has a subtle hint in the answer choices that you could use to avoid some of the work:

The prompt is written as a "probability" question, but since the answers are fractions, you can work backwards and "translate" them into actual values. We're dealing with the first 300 positive integers and asked for the probability of randomly selecting a number that equals an integer raised to a power greater than 1. Here's how the answers can be rewritten:

17/300 = 17 numbers that fit the description
1/15 = 20 numbers
2/25 = 24 numbers
1/10 = 30 numbers
3/25 = 36 numbers

Having your "perfect squares" memorized will make the work go a bit faster; as Brent showed, there are 17 perfect SQUARES (and you SHOULD write them on the pad for easy reference). Finding the perfect cubes won't take too long (but there ARE some values that criss-cross with the perfect squares, so you CAN'T count them twice).

At this point, we have 17 + 4 = 21 values. Working higher (4th power, 5th power, etc.), there cannot be that many additional values that "fit", since we're dealing with a smaller and smaller sub-group each time and we've already seen that there ARE duplicates.

The Number Property that I mentioned earlier is that the "even powers" greater than 2 have all already appeared in your list (as perfect squares)

For example 2�= (2²)(2²) = 4²

So there's no reason to check the 4th, 6th, 8th, etc. powers since there won't be anything new.

Answers A and B are now too small and answers D and E seem way too big. Logically, the answer would have to be 24.

Final Answer: C

GMAT assassins aren't born, they're made,
Rich