The number properties

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nidhis.1408: Master | Next Rank: 500 Posts; Posts: 135; Joined: Mon Oct 03, 2011 6:54 am; Followed by:4 members

The number properties

by nidhis.1408 » Mon Jun 25, 2012 10:30 am

Is k^2 odd?
(1) k - 1 is divisible by 2.
(2) The sum of k consecutive integers is divisible by k.

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Source: Beat The GMAT — Data Sufficiency | Mon Jun 25, 2012 10:30 am

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Mon Jun 25, 2012 10:42 am

nidhis.1408 wrote:Is k^2 odd?

(1) k - 1 is divisible by 2.
(2) The sum of k consecutive integers is divisible by k.

Statement 1: As (k - 1) is divisible by 2, k must be odd.
Hence, kÂ² is also odd.

Sufficient

Statement 2: Say, k consecutive integers are a, (a + 1), (a + 2), ..., and (a + k - 1)
Hence, the sum of these k consecutive integers, S = ak + k(k - 1)/2 = k[a + (k - 1)/2]

Now, if S is divisible by k, then [a + (k - 1)/2] must be integer.
So, (k - 1)/2 must be integer.
So, (k - 1) must be even.

Hence, k must be odd.
Therefore, kÂ² is also odd.

Sufficient

The correct answer is D.

Last edited by Anurag@Gurome on Mon Jun 25, 2012 11:50 am, edited 3 times in total.

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TimeforGMAT: Junior | Next Rank: 30 Posts; Posts: 11; Joined: Tue Aug 31, 2010 5:54 am

by TimeforGMAT » Mon Jun 25, 2012 10:51 am

Dear Anurag,

Can you please explain again how B is sufficient by itself?

Sum of 3(odd) consecutive numbers 1 2 3 is 6 (even) and sum of 2(even) consecutive 1 2 is 3 (odd).

Thank you

Pranith.

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Mon Jun 25, 2012 11:05 am

Thanks for pointing it out.
I've edited my post.

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TimeforGMAT: Junior | Next Rank: 30 Posts; Posts: 11; Joined: Tue Aug 31, 2010 5:54 am

by TimeforGMAT » Mon Jun 25, 2012 11:32 am

Dear Anurag,

I think IMO is A.

Sum of K consecutive integers (K*(K+1)/2) is always divisible by K regardless of K being even or odd.
So we cannot say if K^2 is even or odd.

And, sum of even number of consecutive integers is not always odd (1 + 2 + 3 + 4 = 10)
What am I missing?

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Mon Jun 25, 2012 11:51 am

TimeforGMAT wrote:Sum of K consecutive integers (K*(K+1)/2) is always divisible by K regardless of K being even or odd.
So we cannot say if K^2 is even or odd.

K(K + 1)/2 is the sum of first K integers, which is not same as sum of K consecutive integers.

For the rest part see my edited post above. I've tried to explain it with algebra.