x, y are positive integers. Find the number of even factors of 4x^2

Here is the solution for this question.

Steps 1 & 2: Understand Question and Draw Inferences

We are given that x and y are positive integers. The question asks us to find the number of even factors of \(4x^2\)

The Number of Even Factors of a number = Total Number of Factors – Number of Odd Factors

To find the total number of factors we need to know the powers of each prime factor.

To find the number of odd factors we need to know the powers of every odd prime factor.

So essentially we need to know the prime factorization of \(4*x^2\)

In other words, we need to know the prime factorization of \(x^2\) or \(x\) to be able to answer the given question.


Step 3: Analyze Statement 1: \(x^3 – y^3 + 3xy\) is odd and \(x\) is a prime.

This statement says \(x\) is a prime. So the prime factorization of \(4*x^2\) will be \(2^2 * x^2\)

However, we still don’t know whether \(x\) is even or odd.

If \(x\) is even, then \(x = 2\) which gives us \(4*x^2\) as \(2^4\) and since this expression doesn’t have any odd prime factors, the total number of even factors are same as the total number of factors. (which will be \(5\) in this case.)

However, if \(x\) is an odd prime, then \(4*x^2\) will be \(2^2 * x^2\).

In this case, the total number of factors will be \((2+1)*(2+1) = 9\)

And the number of odd factors will be \((2+1) = 3\)

This gives us the number of even factors = \(9 – 3 = 6\)

So we need to know whether \(x\) is even or odd to determine a unique answer.

We will try to use the other piece of information ” \(x^3 – y^3 + 3xy\) is odd” to determine the even-odd nature of \(x\).

We know that a positive integral power doesn’t affect the even-odd nature. Similarly, multiplication by a odd number won’t affect the even-odd nature.

Therefore if \(x^3 – y^3 + 3xy\) is odd then \(x – y + xy\) must also be odd. So let us now see the various cases where \(x – y + xy\) must be odd.





As you can see, \(x – y + xy\) is odd when \(x\) is odd and also in a case when \(x\) is even.

So we cannot determine whether \(x\) is even or odd.

Therefore statement 1 is not sufficient to arrive at a unique answer.



Step 4: Analyze Statement 2: \(x^{(x+y)}*y^{3x} +x^{3y}\) is odd.

Since positive integral powers do not affect the even-odd nature, statement 2 essentially tells us that \(xy + x\) is odd. Which means \(x*(y+1)\) is odd. So let us now see the various cases where \(x*(y + 1)\) must be odd.




As you can see, “\(x(y + 1)\) is odd” implies that “\(x\) is odd”.

However, \(x\) need not be a prime. In other words, \(x\) could be written as \(x = p^a * q^b * r^c\) … where \(p\), \(q\), \(r\), etc. are primes and \(a\), \(b\), \(c\) are non-negative integers. Therefore without knowing these values we cannot determine the prime factorization of \(x\) and therefore cannot arrive at a unique answer.

Therefore statement 2 is not sufficient.


Step 5: Analyze Both Statements Together (if needed):

From statement 1, we have: \(x\) is a prime and we need to know if \(x\) is even or odd to arrive at a unique answer.

From statement 2, we have: \(x\) is odd.

Therefore both statements together are sufficient.

Answer: Option (C)


Foot Note: To learn more about how to do smarter calculations on Even-Odd questions involving complex terms like the question here, read this article: https://gmatclub.com/forum/do-you-make-these-3-mistakes-in-gmat-even-odd-questions-196654.html

Hope this helps.

Regards,
Krishna