If m and n are positive integers, is n even?
(1) m(m + 2) + 1 = mn
(2) m(m + n) is odd.
OA D
Source: Official Guide
If m and n are positive integers, is n even?
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Let's take each statement one by one.BTGmoderatorDC wrote:If m and n are positive integers, is n even?
(1) m(m + 2) + 1 = mn
(2) m(m + n) is odd.
OA D
Source: Official Guide
(1) m(m + 2) + 1 = mn
Say m = even, thus, mn, the right-hand side = even, thus, m(m + 2) + 1, the left-hand side is even. Or m(m + 2) = Even - 1 = Odd. We know that the product of two odd integers is odd, thus, for m(m + 2) to be odd, m as well as (m + 2) must be odd; however, it invalidates our assumption that m is even. Thus, must be odd.
With m = odd, we have m(m + 2) = Odd and m(m + 2) + 1 = m(m + 2) + Odd = Even. Thus, mn, the right-hand side = Even. Since m is odd, for mn to be even, n must be even. Sufficient.
(2) m(m + n) is odd.
We know that the product of two odd integers is odd, thus, for m(m + n) to be odd, m as well as (m + n) must be odd. This is the same condition that we discussed in Statement 1. Sufficient.
The correct answer: D
Hope this helps!
-Jay
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Statement 1:BTGmoderatorDC wrote:If m and n are positive integers, is n even?
(1) m(m + 2) + 1 = mn
(2) m(m + n) is odd.
Case 1: m is ODD
Plugging m=1 into m(m + 2) + 1 = mn, we get:
1(1+2) + 1 = 1(n)
4 = n
In this case, the answer to the question stem is YES.
Case 2: m is EVEN
Plugging m=2 into m(m + 2) + 1 = mn, we get:
2(2+2) + 1 = 2n
9 = 2n
n = 9/2 = ODD/EVEN
Not viable, since ODD/EVEN = noninteger, violating the condition that n must be an integer.
Since only Case 1 is viable, the answer to the question stem is YES.
Statement 2:
Case 1: m is ODD
Plugging m=1 into m(m + n) is odd, we get:
1(1+n) = ODD
1+ n = ODD
n = ODD - 1 = EVEN
In this case, the answer to the question stem is YES.
Case 2: m is EVEN
Plugging m=2 into m(m + n) is odd, we get:
2(2+n) = ODD
EVENÂ = ODD
Not viable.
Since only Case 1 is viable, the answer to the question stem is YES.
The correct answer is D.
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Followed here and elsewhere by over 1900 test-takers.
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.
For more information, please email me (Mitch Hunt) at [email protected].
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