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For integers x and y, which of the following MUST be an integer?
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19 Feb 2021, 09:31
19 Feb 2021, 09:31
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For integers x and y, which of the following MUST be an integer?
A. \(\sqrt{25x^2+30xy+36y^2}\)
B. \(\sqrt{49x^2−84xy+36y^2}\)
C. \(\sqrt{16x^2−y^2}\)
D. \(\sqrt{64x^2−64xy−64y^2}\)
E. \(\sqrt{81x^2+25xy+16y^2}\)
Kudos for the right answer and explanation
Question part of the project GRE Quantitative Reasoning Daily Challenge - (2021) EDITION
GRE - Math Book
A. \(\sqrt{25x^2+30xy+36y^2}\)
B. \(\sqrt{49x^2−84xy+36y^2}\)
C. \(\sqrt{16x^2−y^2}\)
D. \(\sqrt{64x^2−64xy−64y^2}\)
E. \(\sqrt{81x^2+25xy+16y^2}\)
Kudos for the right answer and explanation
Question part of the project GRE Quantitative Reasoning Daily Challenge - (2021) EDITION
GRE - Math Book
Re: For integers x and y, which of the following MUST be an integer?
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19 Feb 2021, 10:02
19 Feb 2021, 10:02
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1
Hi,
To solve this question, we can ignore the square root for the time being and only focus on the quadratic equation.
Now to find a perfect square it should follow some format like (a+b)^2 or (a-b)^2.
This condition is only satisfied in Option B.
The other thing that you can do is assume x &y as 1.
Try substituting only option B gives us a perfect square whose square root is an integer.
Hope this helps!
To solve this question, we can ignore the square root for the time being and only focus on the quadratic equation.
Now to find a perfect square it should follow some format like (a+b)^2 or (a-b)^2.
This condition is only satisfied in Option B.
The other thing that you can do is assume x &y as 1.
Try substituting only option B gives us a perfect square whose square root is an integer.
Hope this helps!
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Re: For integers x and y, which of the following MUST be an integer?
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19 Feb 2021, 10:08
19 Feb 2021, 10:08
1
Carcass wrote:
For integers x and y, which of the following MUST be an integer?
A. \(\sqrt{25x^2+30xy+36y^2}\)
B. \(\sqrt{49x^2−84xy+36y^2}\)
C. \(\sqrt{16x^2−y^2}\)
D. \(\sqrt{64x^2−64xy−64y^2}\)
E. \(\sqrt{81x^2+25xy+16y^2}\)
A. \(\sqrt{25x^2+30xy+36y^2}\)
B. \(\sqrt{49x^2−84xy+36y^2}\)
C. \(\sqrt{16x^2−y^2}\)
D. \(\sqrt{64x^2−64xy−64y^2}\)
E. \(\sqrt{81x^2+25xy+16y^2}\)
APPROACH #1: Test a pair of values
The question is asking us to determine which expression MUST be an integer for ALL integer values of x and y.
So, let's TEST a pair of values.
Let's plug in x = 1 and y = 1
If an expression evaluates to be a non-integer, we can ELIMINATE that answer choice.
We get...
A)√91.This does NOT evaluate to be an integer. ELIMINATE C
B)√1 = 1. This IS an integer. So, keep B
C)√15. This does NOT evaluate to be an integer. ELIMINATE C
D)√-64. Cannot evaluate. ELIMINATE D
E)√122. This does NOT evaluate to be an integer. ELIMINATE E
By the process of elimination, the correct answer is B
APPROACH #2: Factoring
When we scan the answer choices, we might recognize that answer choice B is in the form of the following special product :
\((a-b)^2 = a^2 - 2ab + b^2\)
Likewise, we have B. \(\sqrt{49x^2−84xy+36y^2}=\sqrt{(7x-6y)^2}=7x-6y\)
Since we're told \(x\) and \(y\) are INTEGERS, we can be certain that \(7x-6y\) is an integer
Answer: B
Cheers,
Brent
