GMAT Number Properties

Question

If k is a positive integer, is k the square of an integer?
(1) k is divisible by 4.
(2) k is divisible by exactly 4 different prime numbers.

From 1) the info is not sufficient as 4 is a square of an integer but 8,12 is not.

Choices left are B, C, and D.

From 2) The info is not sufficient as 210^2 is divisible by exactly 2,3,5,7 which is a square of a number but 210 by itself is not a square of a number.
So with that reasoning you cancel B. Please let me know if this reasoning is not correct.

The choices left are C and E. Now I was not able to find a number which is divisible by 4 and also divisible by exactly 4 different prime numbers.

But whenever I am not able to come to answer for a question from choices C,E the actual answer is always opposite of what I pick

.

My natural hunch was E but I selected C in this case.

But the OA is E.

Did anyone else also comes across this C/E conundrum at all. Any tips on handling the DS problems when it comes to pick choice C or E.

In my case I always have a tendency to pick an incorrect choice.

Thanks.
-A

Ian Stewart · Answer

You got very close to the answer here- your logic is perfect when you look at each statement individually. When it comes to combining the two statements, consider these numbers:

(2^2)*3*5*7 --> this is divisible by 4, and is divisible by exactly four different primes, but is not the square of an integer;

(2^2)*(3^2)*(5^2)*(7^2) --> this is divisible by 4, and is divisible by exactly four different primes, and is the square of an integer; its positive square root is 2*3*5*7.

So we still don't have enough information.

The above illustrates the value of working with prime factorizations- you don't need to bother calculating the numbers here, so it's very fast. You just need to know how to recognize a perfect square from a prime factorization- all the exponents must be even numbers.

aj5105 · Answer

Helpful tip,Ian.

Ian Stewart wrote:You got very close to the answer here- your logic is perfect when you look at each statement individually. When it comes to combining the two statements, consider these numbers:

(2^2)*3*5*7 --> this is divisible by 4, and is divisible by exactly four different primes, but is not the square of an integer;

(2^2)*(3^2)*(5^2)*(7^2) --> this is divisible by 4, and is divisible by exactly four different primes, and is the square of an integer; its positive square root is 2*3*5*7.

So we still don't have enough information.

The above illustrates the value of working with prime factorizations- you don't need to bother calculating the numbers here, so it's very fast. You just need to know how to recognize a perfect square from a prime factorization- all the exponents must be even numbers.

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