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