k = n^2 – 1

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16206; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Jan 22, 2009 8:54 am

If k and n are positive integers, and k = n^2 – 1, which of the following must be true?
I. k is not divisible by 5
II. nk divisible by 3
III. The units digit of k is not 7

(A) III only
(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II and III

Please note that this is not an official GMAT question; it’s my attempt to create difficult (650+ level) GMAT-style questions for this forum.

Brent Hanneson - Creator of GMATPrepNow.com

Source: Beat The GMAT — Problem Solving | Thu Jan 22, 2009 8:54 am

sacx: Master | Next Rank: 500 Posts; Posts: 139; Joined: Wed Oct 22, 2008 4:36 am; Thanked: 17 times

by sacx » Thu Jan 22, 2009 9:03 am

IMO A

the unit digit of n can be from 0 to 9. if you square any of these digits and then subtract 1 you will not get 7 in the units digit

SACX

krisraam: Master | Next Rank: 500 Posts; Posts: 221; Joined: Wed Jan 21, 2009 10:33 amFollowed by: 1 member; Thanked: 12 times

by krisraam » Thu Jan 22, 2009 9:06 am

Answer is D

I. k is not divisible by 5. false. if n is 9. K is divisible by 5. Eliminate B,C,E.
II. nk divisible by 3.
nk = n(n-1)(n+1). These are 3 consecutive numbers. So one of them is 3. so nk is divisible by 3.

by now we know D is the answer by the process of elimination.

If you take 1 to 9. None of their squares has a unit digit 8.

If k and n are positive integers, and k = n^2 – 1, which of the following must be true?
I. k is not divisible by 5
II. nk divisible by 3
III. The units digit of k is not 7

(A) III only
(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II and III

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16206; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Jan 22, 2009 9:08 am

krisraam wrote:Answer is D

I. k is not divisible by 5. false. if n is 9. K is divisible by 5. Eliminate B,C,E.
II. nk divisible by 3.
nk = n(n-1)(n+1). These are 3 consecutive numbers. So one of them is 3. so nk is divisible by 3.

by now we know D is the answer by the process of elimination.

If you take 1 to 9. None of their squares has a unit digit 8.

Nice work, krisraam
The answer is D

If k and n are positive integers, and k = n^2 – 1, which of the following must be true?
I. k is not divisible by 5
II. nk divisible by 3
III. The units digit of k is not 7

(A) III only
(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II and III

Brent Hanneson - Creator of GMATPrepNow.com

gaggleofgirls: Master | Next Rank: 500 Posts; Posts: 138; Joined: Thu Jan 15, 2009 7:52 am; Location: Steamboat Springs, CO; Thanked: 15 times

by gaggleofgirls » Thu Jan 22, 2009 9:33 am

In the case of n=1, then K =0 so nk = 0

So what was keeping me between answer A and answer D was determining if 0/3.

0/3 = 0 Therefore the answer is D.

-Carrie

mdavis: Senior | Next Rank: 100 Posts; Posts: 44; Joined: Thu Nov 20, 2008 2:18 pm; Thanked: 1 times

by mdavis » Thu Jan 22, 2009 10:02 am

How did you figure out that nk=n(n-1)(n+1) ?

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16206; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Jan 22, 2009 10:04 am

mdavis wrote:How did you figure out that nk=n(n-1)(n+1) ?

We know that k = n^2 - 1
We can factor n^2 - 1 to be (n+1)(n-1)
So, k = (n+1)(n-1)
nk = n(n+1)(n-1)

Brent Hanneson - Creator of GMATPrepNow.com

krisraam: Master | Next Rank: 500 Posts; Posts: 221; Joined: Wed Jan 21, 2009 10:33 amFollowed by: 1 member; Thanked: 12 times

by krisraam » Thu Jan 22, 2009 10:29 am

Carie

n can't be 1 as when n is 1 k = n^2 -1 = 0 . K can't be 0 as n,k both are positive integers.

gaggleofgirls: Master | Next Rank: 500 Posts; Posts: 138; Joined: Thu Jan 15, 2009 7:52 am; Location: Steamboat Springs, CO; Thanked: 15 times

by gaggleofgirls » Thu Jan 22, 2009 10:37 am

Thanks for pointing that out, kris.

I noticed that for n to see if n could be 1 but didn't catch that it excludes n=1 since then k is not a pos int.

-Carrie

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