What is the probability that a number chosen between 1 to 15000 both

For a number to have exactly 3 factors, it must be a perfect square of a prime number. So we need to find how many perfect squares of prime numbers are there between 1 to 15,000.

Now, we know that \(12^2 = 144 ; 120^2 = 14400\)
Also, \(13^2 = 169 ; 130^2 = 16900\)

15000 is much closer to 14,400 so we are looking for primes within the first 122/123 positive integers.

We know that there are 25 prime numbers among the first 100 positive integers. (this is something we should know. It helps in approximations).

We also know that a prime can end in 1/3/7/9 only so there are only 4 candidates for primes in every 10 consecutive integers. We usually get about 2-3 primes in such a set.

So from 101 - 122, how many primes are we likely to find? Certainly not 10.
It is fair to say that the total number of primes will be 30 and not 35 so (B) will be the answer, not (A).
All other options are just too big.

A number that has exactly 3 factors is the square of a Prime number. As such, one could look at this problem and say that one has to find out the number of primes such that the square of the prime is less than 15000.

If ONE number is selected at random from 1 to 15000, the total possible outcomes of this experiment = 15000.

Now, it’s good if you can remember that there are 25 primes between 1 and 100. Also, any prime number greater than 7 will always have unit digit as 1 or 3 or 7 or 9. Although the vice versa is not true, this property can be used to find out primes with reasonable accuracy.

Applying this property, there are 8 numbers between 100 and 120 having units digit 1 / 3 / 7 /9. Of these numbers, 111, 117 and 119 are not prime. Therefore, the remaining 5 numbers viz., 101, 103, 107, 109 and 113 are prime.

We do not have to look beyond 120 because \(120^2\) = 14400 and we know that 121 and 122 are not primes. Smart estimation tells us that \(123^2\) will be beyond 15000 and hence we will have to consider numbers between 100 and 120 only.

Therefore, favourable outcomes for the event = 30.

Required probability = \(\frac{30 }{15000}\) = \(\frac{1}{500}\).

The correct answer option is B.

Hope that helps!
Aravind BT

It'll be pretty devilish if i were to give 31/15000 or 4/1875 ( 32/15000) as an option and hide the answer in (none of these ).
But it will only make the question more difficult and less logical

There are 25 Primes under 100 - Noted.

What Bunuel means is: the GMAT will never test if you know how many primes there are less than 122. That's definitely true. So you'd never need to know that if you're taking the GMAT. If instead we're discussing how to answer this particular question, then you certainly need to know, or figure out, how many primes there are less than 122, but the question in this thread is not a realistic GMAT question.

A couple of posters above suggest it's worth knowing there are 25 primes less than 100. I can't imagine the GMAT asking a question where that knowledge would be helpful, and I'd be curious if anyone can point to a single official GMAT question where you'd benefit from knowing how many primes there are less than 100.

Thanks for clarifying. The GMAT seems to have a lot of practice questions out there that don't conform to the way the test challenges test takers. From most official answers I've read, the difficulty is in determining the approach, the underlying relations, what they're really asking, or how we could go about testing something. A lot of practice questions miss the mark by making the GMAT more about algebra or brute force than finesse and reasoning. Still helpful to know I won't see something like this come test day.

--

On topic I also considered another approach for this question before I knew I wasn't likely to come across it again. This is without the 25 primes in 1-100 knowledge.

-> 3 factors therefore only counting squares of primes.
-> sqrt(15000) = sqrt(100 150) = sqrt(100 25 6) = 50sqrt(6). To test for sqrt 6 I considered sqrt(600). 24 shoots under (144*4 = 576) and 25 shoots over (5^4 = 625). It's almost exactly half so 24.5 should be a good approximation. Which makes 2.45 close to sqrt(6) value.

50*2.45 = 122.5 So we're looking for all the primes up to 122.

I first take out 1 (-1) as we won't catch it later. Then I take out non prime numbers with factors of 2 or 3.
122/2 v = 61. But the original 2 is a prime so it's (-60)
122/3 v = 40. But the original 3 is a prime so it's (-39)
Add back sixes as we removed these twice. 122/6 = (+20)

We could do the same for say 5, but it would get complicated. Ex: We'd end up removing 90 again by removing multiples of 15. It gets too messy to add back sub groups. Instead we can note any multiple of a larger prime has not been removed if that multiple is also a prime higher than 2 or 3.

That is to say 5* 2/3/4/6/8/9/10 etc is accounted for. but 5 * prime that isn't 2/3 has not been accounted for.

So let's remove what we haven't handled. 122/5 v = 24 so we are looking for primes up to 24.

5*5, 5*7, 5*11, 5*13, 5*17, 5*19, 5*23. Don't need to solve, we know they aren't double counted.

Same logic for larger primes. 122/7 v = 17. 122/11 v = 11. 122/13 = 9 (9 is the highest factor that fits, everything has been accounted for)

7*7, 7*11, 7*13, 7*17

11*11

(-12)

122 - 1 - 60 - 39 + 20 - 12 = 30

Probably not worth brute forcing, but anyways it's a different way to approach I guess.

Posted from my mobile device

You actually just invented from scratch one of the most famous prime-counting methods from early Number Theory - in general, those methods are called "sieve methods", and the particular one you're describing is known as the Sieve of Eratosthenes. I just glanced at the wikipedia page to provide a link in case you're curious about it, and serendipitously it has a graphic demonstrating how the sieve applies when finding all primes up to 121, which is exactly what the question in this thread asks about.

https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

That sieve is never going to be useful on the GMAT, but if you can figure out on your own some of the more important foundations of classical number theory, you shouldn't have much difficulty with the test!

Super cool! Thanks for the kind words and the link - it made for an interesting read. The thing I've enjoyed the most about studying for the GMAT is number properties - especially stuff about primes I'm finding it really interesting. I feel like I'm starting to learn Math at a much more intuitive level just from understanding the basic integer building blocks. My school system was mostly just memorization and answering exactly the questions we were taught to solve.

I appreciate the vote of confidence, but the places I imagine I'm more likely to lose questions now is stuff like absolutes, exponents/roots, and geometry. Still got quite a journey ahead of me.

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