Number Systems -Odds & Evens

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sukhman: Master | Next Rank: 500 Posts; Posts: 202; Joined: Sun Sep 08, 2013 11:51 amFollowed by: 2 members; Thanked: 3 times

by sukhman » Wed Oct 16, 2013 9:41 am

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

Last edited by sukhman on Wed Oct 16, 2013 10:59 pm, edited 1 time in total.

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Source: Beat The GMAT — Data Sufficiency | Wed Oct 16, 2013 9:41 am

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Wed Oct 16, 2013 11:36 am

sukhman wrote:If a and b are both positive integers, is ba+1 - bab odd?
(1) a + (a + 4) + (a - 8) + (a + 6) + (a - 10) is odd
(2) b3 + 3b2 + 5b + 7 is odd

Are you sure you transcribed this question correctly?
For example, it seems peculiar to write bab
Is there an exponent hiding here?

I ask because you seem reluctant to check your posts to ensure that any exponents are clearly evident (e.g., b3 + 3b2)

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Brent

Brent Hanneson - Creator of GMATPrepNow.com

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mainbhidhruv: Newbie | Next Rank: 10 Posts; Posts: 6; Joined: Tue Oct 29, 2013 6:46 am

by mainbhidhruv » Tue Oct 29, 2013 6:55 am

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 !

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Tue Oct 29, 2013 7:18 am

mainbhidhruv 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

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 !

EDIT: I originally agreed with you, but mevicks' awesome solution showed me otherwise.
The correct answer is D

Great work, mevicks!

Cheers,
Brent

Last edited by Brent@GMATPrepNow on Tue Oct 29, 2013 7:47 am, edited 1 time in total.

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mevicks: Master | Next Rank: 500 Posts; Posts: 269; Joined: Thu Sep 19, 2013 12:46 amFollowed by: 7 members; Thanked: 94 times

by mevicks » Tue Oct 29, 2013 7:42 am

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 Â± 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

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theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pmFollowed by: 34 members; Thanked: 448 times; GMAT Score:650

by theCodeToGMAT » Tue Oct 29, 2013 7:51 am

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

R A H U L

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Tue Oct 29, 2013 9:02 am

I thought I'd mention that, for these kinds of questions, it's useful to create a table that allows you to test various cases (e.g., a is even and b is even, or a is odd and b is even, etc)

If anyone is interested, we have a free video on this topic: https://www.gmatprepnow.com/module/gmat- ... ies?id=839

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Wed Oct 30, 2013 11:04 pm

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 * báµƒ - b * aáµ‡ odd?

Start by factoring out b and rewriting the question as "Is b * (báµƒ - aáµ‡) odd?"

Now let's thinking about what we know. b is clearly an integer, and (báµƒ - aáµ‡) 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 (báµƒ - aáµ‡) 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 aáµ‡ is also odd, as it's just a bunch of odd numbers multiplied together.

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

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

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

S2::

bÂ³ + 3bÂ² + 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 (báµƒ - aáµ‡) are odd. SUFFICIENT!

Good q, very 2013 GMAT.

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