If n and k are integers and n^2 – kn is even, which of the following

Question

If n and k are integers and n^2 – kn is even, which of the following must be even?

A) n^2
B) k^2
C) 2n + k^2
D) n(k + 1)
E) k(k + n)

BrentGMATPrepNow · Accepted Answer

A lot of Integer Properties questions can be answered quickly by  testing values  that satisfy the given information.

So, for example, if n² - kn is EVEN, it COULD be the case that n = 1 and k = 1
Now use these values to check the answer choices
A) n² = 1² = 1 (not even - ELIMINATE)
B) k² = 1² = 1 (not even - ELIMINATE)
C) 2n + k² = 2(1) + 1² = 3 (not even - ELIMINATE)
D) n(k + 1) = 1(1 + 1) = 2 (EVEN - keep!!)
E) k(k + n) = 1(1 + 1) = 2 (EVEN - keep!!)

We are now down to answer choices D and E. 
So let's test a second pair of values.
If n² - kn is EVEN, it could also be the case that n = 2 and k = 1
Now use these values to check the remaining answer choices
D) n(k + 1) = 2(1 + 1) = 4 (EVEN - keep!!)
E) k(k + n) = 1(1 + 2) = 3 (not even - ELIMINATE)

By the process of elimination the answer must be D

Cheers,
Brent

danielemondi1 · Answer

You can easily test cases by assuming if n and k are even or odd

n*n - k*n = Even

Case #1

n = Odd
k = Odd

Odd * Odd - Odd * Odd = Even

This holds since we know that:

Odd*Odd = Odd
Even + Odd = Odd

Now you can eliminate
(A)
(B)
(C)
Since those would be all odds by the same conditions

So you are left with (D) and (E), which are both Even

now you can assume

n = Even
k = Odd

Even*Even - Even*Odd = Even ?

Yes by:

Even*Even =Even 
Even*Odd = Even
Even + Even = Even

Therefore we test again our two answer choices left:

(D) Even(Odd+Odd) = Even
(E) Odd(Odd+Even) = Odd

so (D) is correct!

sthahvi · Answer

can you explain why did we take the first case for 1,2,3 and second for the last two ?

Fdambro294 · Answer

N^2 - KN = Even Integer (and both K and N are Integers

N * (N - K) = Even Integer

Case 1:

N = odd

Then (N - K) must be Even ——- (odd - K) = Even

K must = Odd

Case 2:

N = Even

Then (N - K) can be either Even or Odd

(Even - K) = even ———— K = Even

Or

(Even - K) = odd ———— K = Odd

Summary: we have 3 possibilities:

(1) N = odd -and- K = Odd

(2) N = Even -and- K = Even

(3) N= Even -and- K = Odd

Only answer (D) must be true given all 3 possible cases

N(K + 1)

Possibility: N = odd ——-> (k + 1) = odd + odd = even

Even result.

Possibilities 2 and 3: Factor N is Even —-> so the product must be an Even result

Posted from my mobile device

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If n and k are integers and n^2 – kn is even, which of the following

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If n and k are integers and n^2 – kn is even, which of the following

Prep Toolkit

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