m, n and k are positive integers. If the

Question

m, n and k are positive integers. If the product mn is odd, is k odd?

(1) mn + n + k is odd
(2) n^2 – kn – 6k^2 is even

*Kudos for all correct solutions

niks18 · Answer

m*n=Odd  implies that  m=Odd  &  n=Odd

Statement 1:   mn+n+k=odd

=> Odd+Odd+k=Odd => k=Odd .  Sufficient

Statement 2:   n^2-kn-6k^2=Even

=> Odd-k*Odd-Even=Even

Therefore  k*Odd=Odd => k=Odd .  Sufficient

Option  D

HKD1710 · Answer

m*n is odd it means both m and n has to be odd.

stmt-1:  mn + n + k = odd 
as we know m and n is odd then we have
odd + odd + k = odd
even + k = odd 
now k has to be odd here. so  sufficient .

stmt-2: 
n^2 -kn - 6k^2 = even
n^2 -k(n+6k) = even
n^2 - even = k(n+6k) 
as we know n is odd, n^2 is also odd, so
odd - even = k(n+6k) 
odd = k(n+6k) (because odd - even = odd)

for k*(n+6k) to be odd both k and n+6k has to be odd. so we know k is odd. 
 Sufficient.

BrentGMATPrepNow · Answer

Target question :    Is k odd?

Given : The product mn is odd  
If the product mn is odd, when we know that  m is ODD  and  n is ODD

Statement 1 : mn + n + k is odd 
In other words: ( ODD )( ODD ) +  ODD  + k is odd
Simplify: ODD + ODD + k is odd
Simplify more: EVEN + k is odd
This means  k must be ODD 
So, the answer to the target question is  YES, k IS odd 
Since we can answer the  target question  with certainty, statement 1 is SUFFICIENT

Statement 2 : n² – kn – 6k² is even 
Let's factor the expression to get: (n - 3k)(n - 2k) is even
In other words: ( ODD  - 3k)( ODD  - 2k) is even
Notice that 2k must be EVEN, so we can write: ( ODD  - 3k)( ODD  - EVEN) is even
ODD - EVEN = ODD, so we can now say: ( ODD  - 3k)(ODD) is even
This tells us that ( ODD  - 3k) must be EVEN
In order for ( ODD  - 3k) to be EVEN, it must be the case that 3k is ODD
If 3k is ODD, then  k must be ODD 
So, the answer to the target question is  YES, k IS odd 
Since we can answer the  target question  with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent

unraveled · Answer

m, n and k are positive integers. If the product mn is odd, is k odd?

(1) mn + n + k is odd
odd + odd + k = odd
k = even - odd
k is odd

SUFFICIENT.

(2) n^2 – kn – 6k^2 is even
odd - k*odd - even*k*k = even
odd - odd = even

Therefore 
k*odd + even*k*k = odd
It is only possible when k is odd.

SUFFICIENT

Answer D.

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