a data from gmat club

[diebeatsthegmat]

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

what does "the integer so chosen equals an integer raised to an exponent that is an integer greater than 1?" mean? i dont understand this

[smackmartine]

Options do not seem to be correct.

Here is my approach.

Question says N=n^p , where p>=2 (where N and n are integers).

Considering squares:
17^2 = 289 , so after 17 all the squares will be greater than 300.
# of favorable outcomes is 17

Considering cubes:
6^3 is 216 ,so after 6 all the cubes will be greater than 300.
As we have already considered 1 (any higher power will result in 1), so # of favorable outcomes is 5 (2^3 through 6^3)

we should skip considering 4th powers because these are also perfect squares, which we have already considered.

Considering 5th powers:

3^5 < 300 ,so # of favorable outcomes is 2 (2^5 and 3^5)

we should skip considering 6th powers because these are also perfect squares, which we have already considered.

Considering 7th powers:

2^7 < 300 ,so # of favorable outcome is 1

Any higher powers of 2 will result in integer greater than 300. SO STOP !!

Adding total # of favorable outcomes , we get 17+5+2+1 =25

Also , total # of outcomes is 300

So, the probability that the integer so chosen equals an integer raised to an exponent that is an integer greater than 1

= 25/300 = 1/12

Experts please comment!!!

[lunarpower]

the above approach has only one flaw: it fails to recognize that 4^3 = 64, which has already been counted as 8^2; therefore, you've over-counted the # of cubes by 1.

so looks like you should have not 25/300 but 24/300.

[lunarpower]

what does "the integer so chosen equals an integer raised to an exponent that is an integer greater than 1?" mean? i dont understand this

let the integer you pick = x.
then, rephrase:
"what is the probability that x = m^n, where m and n are integers and n > 1?"

components of the rephrase:
"the integer so chosen" = x
"an integer" = m
"power that is an integer > 1" = n

does that help?

[smackmartine]

True.I missed 4^3 .Thanks Ron.

[smackmartine]

so looks like you should have not 25/200 but 24/200.

Typo 24/300

[diebeatsthegmat]

what does "the integer so chosen equals an integer raised to an exponent that is an integer greater than 1?" mean? i dont understand this

let the integer you pick = x.
then, rephrase:
"what is the probability that x = m^n, where m and n are integers and n > 1?"

components of the rephrase:
"the integer so chosen" = x
"an integer" = m
"power that is an integer > 1" = n

does that help?

so does it mean that whether the chosen number x is equal to A^a with a>1 and A^a and X belong numbers from 1 to 300 inclusive?

[lunarpower]

so does it mean that whether the chosen number x is equal to A^a with a>1 and A^a and X belong numbers from 1 to 300 inclusive?

basically, yes (provided that "capital A" and "lowercase a" are two different variables -- the choice of 2 different forms of the same letter is not ideal here).

the problem does not explicitly stipulate that "A" or "a" must be 300 or less, but that is a necessary consequence of the fact that X must be 300 or less.

[uslalas22]

Is there a quick method besides going through all of the exponents one by one as the solution here proposes. Or its just a matter of knowing these exponential properties?

[lunarpower]

Is there a quick method besides going through all of the exponents one by one as the solution here proposes. Or its just a matter of knowing these exponential properties?

no, not really a "quick method"; in fact, a big part of the purpose of problems like this one is to foil people who are looking for quick algebraic methods for everything.

if you don't know these numbers already:
you should make ARITHMETIC FLASH CARDS with the following numbers:

* perfect squares, up to 25^2 (i.e., 11^2, 12^2, 13^2, ..., 25^2)

* perfect cubes, up to 10^3 (i.e., 1^3, 2^3, 3^3, ..., 10^3)

* powers of 2, up to 2^10 (i.e., 2^1, 2^2, 2^3, ..., 2^10)
- in addition to knowing the exact values (up to 2^10 = 1024), you should also know that 2^10 is "approximately 1000". that will help a lot with estimation.

* powers of 3, up to 3^6 (i.e., 3^1, 3^2, 3^3, ..., 3^6)

* powers of 10, up to 10^9 (i.e., 10^1, 10^2, 10^3, ..., 10^9)

mix these all up and quiz yourself.

[sanju09]

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

what does "the integer so chosen equals an integer raised to an exponent that is an integer greater than 1?" mean? i dont understand this

This means that a number so chosen can be written in the form a^n where a is an integer such that 1 ≤ a ≤ 300 and n is an integer such that n > 1.


I am rewriting the answer choices

(A) 17/300
(B) 20/300
(C) 24/300
(D) 30/300
(E) 36/300

In other words, how many integers between 1 and 300 are integers raised to an exponent that is an integer greater than 1?
(A) 17
(B) 20
(C) 24
(D) 30
(E) 36

And the Quick List is

1 = 1^ (an integer greater than 1)

4 = 2^2

8 = 2^3

9 = 3^2

16 = 2^4 = 4^2

25 = 5^2

27 = 3^3

32 = 2^5

36 = 6^2

49 = 7^2

64 = 2^6 = 4^3 = 8^2

81 = 3^4 = 9^2

100 = 10^2

121 = 11^2

125 = 5^3

128 = 2^7

144 = 12^2

169 = 13^2

196 = 14^2

216 = 6^3

225 = 15^2

243 = 3^5

256 = 2^8 = 4^4 = 16^2

289 = 17^2

I can count [spoiler]24


C[/spoiler]

[diebeatsthegmat]

so does it mean that whether the chosen number x is equal to A^a with a>1 and A^a and X belong numbers from 1 to 300 inclusive?

basically, yes (provided that "capital A" and "lowercase a" are two different variables -- the choice of 2 different forms of the same letter is not ideal here).

the problem does not explicitly stipulate that "A" or "a" must be 300 or less, but that is a necessary consequence of the fact that X must be 300 or less.

thank you indeed... sometimes english makes me crazzziiiii