Bunuel wrote:
kirankp 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
Basically we need to find how many m^n (where n>1) are between 1 and 300 inclusive.
For n=2 --> m^2<300 --> m<18, so there are
17 such numbers: 1^2=1, 2^2=4, 3^2=9, 4^2=16, ..., 17^2=289;
For n=3 --> m^3<300 --> m<7, so there are 6 such numbers:
1^3=1, 2^3=8, 3^3=27,
4^3=64, 5^3=125, 6^3=216. 1^3=1=1^2 and 4^3=64=8^2 have already been counted so, that leaves only
4 numbers;
Skip n=4, since all perfect fourth power numbers are also perfect squares;
For n=5 --> m^5<300 --> m<4, so there are 3 such numbers:
1^5=1, 2^5=32, 3^5=243. 1^5=1=1^2 has already been counted so, that leaves only
2 numbers;
Skip n=6 for the same reason as n=3;
For n=7 --> m^7<300 --> m<3, so there are 3 such numbers:
1^7=1, 2^7=128. 1^7=1=1^2 has already been counted so, that leaves only
1 numbers.
Total: 17+4+2+1=24.
The probably thus equals to 24/300=2/25.
Answer: C.
Bunuel,
I tried counting it differently.
Instead of chopping it up according to degree of the number, I went by the numbers themselves:
2 : n can be - 1,2,3,5,6,7 . 3 : n can be - 1,2,3,5 . 5 : n can be - 1,2,3 . for the numbers 6,7,10,11,12,13,14,15,16,17 n can be 2.
so we have 5+4+3+10*2 = 31 and not 24 like you got.
Why am I getting a different result?
Can you see what I'm counting twice?