If n^2/n yields an integer greater than 0, is n divisible by 30?

Given: n^2/n yields an integer greater than 0 --> n>0 --> n^2/n=n=integer. So, the question basically asks whether n is a multiple of 30.

(1) n^2 is divisible by 20 --> 20=2^2*5 --> n is divisible by at least 2 and 5, otherwise these factors could not appear in n^2 (exponentiation does not "produce" primes), though we don't know about other possible factors of n. Not sufficient.

(2) n^3 is divisible by 12 --> 12=2^2*3 --> the same way here: n is divisible by at least 2 and 3, though we don't know about other possible factors of n. Not sufficient.

(1)+(2) n is divisible by 2, 3 and 5 hence by 2*3*5=30. Sufficient.

Answer: C.
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Hi,

If n*n is divisible by 20 (2*2*5) then in the numerator there has to be minimum 2*2*5*5 so as to make it a perfect square and that 20 divides the numerator properly and gives us an integer. So n has atleast one 2 and one 5.

Similarly, n*n*n = n^3 is divisible by 12 (2*2*3). So numerator has to have 2*2*2*3*3*3 so as to make it a perfect cube and that 12 divides the numerator and gives us an integer value. So n has atleast one 2 and one 3

Therefore combining the 2 statements n has atleast one 2, one 3 and one 5. Hence it should be divisible by 30.

Consider Kudos if the post helps!!

Hi bunuel!

I have a small doubt. In your solution; 2,5 and 2,3 are factors of n^2 and n^3 respectively. So how do they become factors of n as well ? Can you please explain this concept in more detail ?

Thanks

Ask yourself if n is an integer and n^2 is a multiple of 2 can n NOT be a multiple of 2? If n is NOT a multiple of 2, how can this prime "appear" in n^2?

As said in my post above: exponentiation does not "produce" primes.
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Hi Bunuel

Going by the above theory ( exponentiation does not "produce" primes ) i approached a question on one of GMAT Club flash cards -:
Is integer x^2*y^4 divisible by 9?

1) x is an integer divisible by 3
2) xy is an integer divisible by 9

Its answer is E.

But i think B(Statement 2) is sufficient. exponentiation does not "produce" primes

What am i missing here?

That question is discussed here: is-x-2-y-4-an-integer-divisible-by-100947.html
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