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If x and y are prime numbers, how many factors has x^2y^2?

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MathRevolution
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If x and y are prime numbers, how many factors has x^2y^2? [#permalink] New post   21 Aug 2018, 03:36
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[Math Revolution GMAT math practice question]

If \(x\) and \(y\) are prime numbers, how many factors has \(x^2y^2\)?

1) \(xy=10\)
2) \(x+y\) is odd
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D
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Re: If x and y are prime numbers, how many factors has x^2y^2? [#permalink] New post   21 Aug 2018, 05:13
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To count the factors of a positive integer:
1. Prime-factorize the integer
2. Write the prime-factorization in the form (a^p)(b^q)(c^r)...
3. The number of factors = (p+1)(q+1)(r+1)...

MathRevolution wrote:
[Math Revolution GMAT math practice question]

If \(x\) and \(y\) are prime numbers, how many factors has \(x^2y^2\)?

1) \(xy=10\)
2) \(x+y\) is odd


Only two cases are possible:
Case 1: x≠y
Here, to determine the number of factors for \(x^2y^2\), we add 1 to each exponent and multiply:
(2+1)(2+1) = 9 factors
Case 2: x=y, with the result that \(x^2y^2 = x^4\)
Here, to determine the number of factors for \(x^4\), we add 1 to the exponent:
4+1 = 5 factors

Implication:
To determine the number of factors, we need to know whether x=y.

Statement 1:
Only one pair of prime numbers has a product of 10:
2 and 5.
Thus, x≠y.
SUFFICIENT.

Statement 2:
Since x+y = odd, either x or y is ODD, while the other value is EVEN.
Thus, x≠y.
SUFFICIENT.

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Re: If x and y are prime numbers, how many factors has x^2y^2? [#permalink] New post   21 Aug 2018, 03:42
From statement 1: xy = 10. Only poasible combination of x and y is (2,5) or (5,2). This is sufficient.

From statement 2: x+y = odd. Here also only possible combination is either x or y should be even. X or Y = 2.Hence sufficient.

D is the answer

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If x and y are prime numbers, how many factors has x^2y^2? [#permalink] New post   21 Aug 2018, 05:06
MathRevolution wrote:
[Math Revolution GMAT math practice question]

If \(x\) and \(y\) are prime numbers, how many factors has \(x^2y^2\)?

1) \(xy=10\)
2) \(x+y\) is odd



Note: x and y are prime numbers.

We are looking for the number of factors of \(x^2y^2\).

Total number of factors = (2+1) (2+1) = 3*3 = 9.

is this question okay ? or i am wrong.
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Re: If x and y are prime numbers, how many factors has x^2y^2? [#permalink] New post   23 Aug 2018, 02:10
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=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

If \(x\) and \(y\) are different prime numbers, then \(xy\) has \((2+1)(2+1) = 9\) factors.
If \(x\) and \(y\) are the same prime number, then \(xy\) has \(4+1 = 5\) factors.

Condition 1)
Since \(x\) and \(y\) are prime numbers and \(xy = 10\), either \(x = 2\) and \(y = 5,\) or \(x = 5\) and \(y = 2.\)
So, \(x\) and \(y\) are different prime numbers. Thus, condition 1) is sufficient.

Condition 2)
Since \(x\) and \(y\) are prime numbers and \(x + y\) is odd, one of them is even and the other one is odd.
So, \(x\) and \(y\) are different prime numbers. Thus, condition 2) is sufficient.

Therefore, D is the answer.

Answer: D
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Re: If x and y are prime numbers, how many factors has x^2y^2? [#permalink] New post   30 Aug 2018, 23:19
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selim wrote:
MathRevolution wrote:
[Math Revolution GMAT math practice question]

If \(x\) and \(y\) are prime numbers, how many factors has \(x^2y^2\)?

1) \(xy=10\)
2) \(x+y\) is odd



Note: x and y are prime numbers.

We are looking for the number of factors of \(x^2y^2\).

Total number of factors = (2+1) (2+1) = 3*3 = 9.

is this question okay ? or i am wrong.


That is right.
The point of this question is that x and y are different prime number or not.
From each of conditions, we can conclude x and y are different prime numbers, and the number of factors is (2+1)(2+1) = 3.
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Re: If x and y are prime numbers, how many factors has x^2y^2? [#permalink] New post   30 Aug 2018, 23:39
MathRevolution wrote:
[Math Revolution GMAT math practice question]

If \(x\) and \(y\) are prime numbers, how many factors has \(x^2y^2\)?

1) \(xy=10\)
2) \(x+y\) is odd



Good Question!

x and y are primes - could be 2 and any other prime number or any other prime pair (apart from 2)

1. xy =10 ; possibility - 2 and 5 (sufficient)

2. x + y = odd -- here again the possiblilty is 2 and any other number ( sufficient)

D
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Re: If x and y are prime numbers, how many factors has x^2y^2? [#permalink] New post   08 Sep 2023, 15:29
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Re: If x and y are prime numbers, how many factors has x^2y^2? [#permalink]
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