If a and b are both positive integers, is b^(a+1) - ba^b odd?
(1) a + (a + 4) + (a - 8) + (a + 6) + (a - 10) is odd
(2) b^3 + 3b^2 + 5b + 7 is odd

b^(a+1)-ba^b = b(b^a-a^b)

I. 5a-4 no reference of b - not sufficient
II. b^3 + 3b^2 + 5b + 7 is odd for all the even values of b and will make b(b^a-a^b) even
Sufficient.

OA B

Brent, Please correct me if I am wrong here !

sukhman wrote:If a and b are both positive integers, is b^(a+1) - ba^b odd?
(1) a + (a + 4) + (a - 8) + (a + 6) + (a - 10) is odd
(2) b^3 + 3b^2 + 5b + 7 is odd

Umm, I think the answer should be D
Here is my reasoning:

Note:
Odd ± Even = Odd
Odd * Odd = Odd


Q: b^(a+1) - ba^b = odd?

St1:
a + (a + 4) + (a - 8) + (a + 6) + (a - 10) = odd
5a - 8 = odd
a = odd

If b = odd ---> b^(a+1) - ba^b = odd^even - odd*odd^odd = odd - odd = even
If b = even --> b^(a+1) - ba^b = even^even - even*odd^even = even
Thus b^(a+1) - ba^b is not odd.
SUFFICIENT

St2:
b^3 + 3b^2 + 5b + 7 = odd
(b^3 + 3b^2 + 5b) + 7 = odd
(b^3 + 3b^2 + 5b) = even

If b = odd ---> (b^3 + 3b^2 + 5b) = odd + odd + odd = odd ---> INVALID from the above deduction
If b = even --> Is the only VALID option

Now, original equation : b^(a+1) - ba^b with b = even becomes
Even - even = Even
Thus b^(a+1) - ba^b is not odd
SUFFICIENT

Answer D

Yes, Answer must be [spoiler]{D} [/spoiler]

To find: (b^a)(b) - (b)(a^b) --> odd

b[ b^a - a^b ]

Statement 1:
a + (a + 4) + (a - 8) + (a + 6) + (a - 10) --> ODD
5a -8 = ODD
5A = ODD
A = ODD

If b= ODD
ODD [ ODD^ODD - ODD^ODD] => ODD [EVEN] = EVEN

If b= EVEn
EVEn {..} --> Always EVEN
SUFFICIENT

Nice question! Just want to sum up the above (and make it a little more legible).

OG question: If a and b are positive integers, is b * bᵃ - b * aᵇ odd?

Start by factoring out b and rewriting the question as "Is b * (bᵃ - aᵇ) odd?"

Now let's thinking about what we know. b is clearly an integer, and (bᵃ - aᵇ) is an integer minus an integer, so it's an integer too. If the product of two integers is odd, then BOTH integers being multiplied must be odd themselves, so our question is really

"Are both b and (bᵃ - aᵇ) odd?"

Now the statements should be a lot easier to work with.

S1:: 5a - 8 is odd, so a is odd.

If a is odd, then aᵇ is also odd, as it's just a bunch of odd numbers multiplied together.

If b is odd, then bᵃ is also odd, so (bᵃ - aᵇ) = (odd - odd) = even, and the answer to our question is NO, it is NOT the case that both b and (bᵃ - aᵇ) are odd.

If b is even, then clearly the answer again is NO, it is NOT the case that both b and (bᵃ - aᵇ) are odd.

So in either case, SUFFICIENT - one of the numbers is even!

S2::

b³ + 3b² + 5b + 7 is odd. If b is odd, we have (Odd + Odd) + (Odd + Odd) = (Even + Even) = Even ... but our sum is supposed to be odd, per the prompt, so b is even.

If b is even, then it is NOT the case that both b and (bᵃ - aᵇ) are odd. SUFFICIENT!

Good q, very 2013 GMAT.