If x and y are positive integers, is x^2*y^2 even ?
(1) x + 5 is a prime number
(2) y + 1 is a prime number
OA A
Source: Magoosh
If x and y are positive integers, is x^2*y^2 even ?
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Given: x and y are positive integers.BTGmoderatorDC wrote:If x and y are positive integers, is x^2*y^2 even?
(1) x + 5 is a prime number
(2) y + 1 is a prime number
OA A
Source: Magoosh
Question: Is x^2*y^2 even?
For x^2*y^2 to be even, at least one between x and y must be even.
Question rephrased: Is at least one between x and y even?
Let's take each statement one by one.
(1) x + 5 is a prime number.
=> (x + 5) must be a prime number, greater than 5. Since all the prime numbers greater than 5 are odd.
=> x + 5 = Odd => x + Odd = Odd => x = Odd - Odd => x = Even. Sufficient.
(2) (y + 1) is a prime number.
Say, y + 1 = 2 => y = 1 (Odd). If x = Odd, then the answer is No; however, if x = Even, the answer is Yes. No unique answer. Insufficient.
The correct answer: A
Hope this helps!
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$$x,y\,\, \geqslant 1\,\,\,{\text{ints}}\,\,\,\,\left( * \right)$$BTGmoderatorDC wrote:If x and y are positive integers, is x^2*y^2 even ?
(1) x + 5 is a prime number
(2) y + 1 is a prime number
Source: Magoosh
$${\left( {xy} \right)^2}\,\,\mathop = \limits^? \,\,\,{\text{even}}\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\boxed{\,\,?\,\,\,:\,\,x\,\,{\text{even}}\,\,\,{\text{or}}\,\,\,y\,\,{\text{even}}\,\,}$$
$$\left( 1 \right)\,\,\left\{ \matrix{
x + 5\,\,\,\,\mathop \ge \limits^{\left( * \right)} \,\,\,6 \hfill \cr
x + 5\,\,{\rm{prime}} \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,x + 5\,\, = {\rm{odd}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,x\,\,{\rm{even}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
$$\left( 2 \right)\,\,\,y + 1\,\,{\rm{prime}}\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {2,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
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x²y² will be even unless both x and y are ODD.BTGmoderatorDC wrote:If x and y are positive integers, is x^2*y^2 even ?
(1) x + 5 is a prime number
(2) y + 1 is a prime number
Question stem, rephrased:
Are x and y both odd?
Statement 1:
x+5 is prime number greater than 2 and thus must be ODD.
Implication:
x+5 = ODD
x = ODD - 5 = ODD - ODD = EVEN.
Since x is even, x and y cannot both be odd.
SUFFICIENT.
Statement 2:
Here, it's possible that y=1, with the result that y+1 = 1+1 = 2 = prime.
If y=1 and x=1, then x and y are both odd.
If y=1 and x=2, then x and y are NOT both odd.
INSUFFICIENT.
The correct answer is A.
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I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
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