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# If n is a positive integer and r is the remainder when

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Manager
Joined: 04 Jun 2008
Posts: 141
If n is a positive integer and r is the remainder when [#permalink]

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15 Aug 2008, 10:54
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If n is a positive integer and r is the remainder when (n-1)(n+1) is divided by 24, what is
the value of r?
(1) 2 is not a factor of n.
(2) 3 is not a factor of n.
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is
sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

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Joined: 07 Nov 2007
Posts: 1733
Location: New York

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Updated on: 18 Aug 2008, 07:59
1
chan4312 wrote:
If n is a positive integer and r is the remainder when (n-1)(n+1) is divided by 24, what is
the value of r?
(1) 2 is not a factor of n.
(2) 3 is not a factor of n.
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is
sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

1) 2 is not factor n.
n can be any odd number.
say n=3 n-1 * n+1 = 2*4=8 remainder =8
say n=5 n-1 * n+1 = 4*6=24 remainder =0

not suffcient
2) 3 is not factor of n.
say n=2 n-1 * n+1 = 1*3=3 remainder =3
say n=4 n-1 * n+1 = 3*5=15 remainder =15
nof sufficient

combine both

say n=5 n-1 * n+1 = 6*8=48 remainder =0
say n=7 n-1 * n+1 = 8*10=80 remainder =8

not sufficient.

E.

I did silly mistake again.
n=5 n-1 * n+1 = 4*6=24 remainder =0
n=7 n-1 * n+1 = 6*8=48 remainder =0

Agree with C.
_________________

Smiling wins more friends than frowning

Originally posted by x2suresh on 15 Aug 2008, 11:59.
Last edited by x2suresh on 18 Aug 2008, 07:59, edited 2 times in total.
Intern
Joined: 02 Jul 2007
Posts: 3

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15 Aug 2008, 13:09
(C)

Using (1), if 2 is not a factor of n then -
For n=1, r=0
For n>1, (n-1) and (n+1) are two consecutive EVEN integers. Hence, the product (n-1)(n+1) is always divisible by 8 but tells us nothing about their divisibility by 24!

Using (2), if 3 is not a factor of n then -
For n=1, r=0
For n>1, either (n-1) or (n+1) is divisible by 3 [any number which is NOT divisible by 3 can be written as 3x-2, or 3x-1, or 3x+1, or 3x+2; you can see why either (n+1) or (n-1) is divisible by 3]. This too does not tell us anything about their divisibility by 24!

Using (1) and (2) together,
For n=1, r=0
For n>1, the product (n-1)(n+1) must be divisible by 8 as well as by 3. Hence r=0.
Senior Manager
Joined: 19 Mar 2008
Posts: 344

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18 Aug 2008, 09:00
If n is a positive integer and r is the remainder when (n-1)(n+1) is divided by 24, what is
the value of r?
(1) 2 is not a factor of n.
(2) 3 is not a factor of n.

(1)
if 2 is not a factor of n, one of (n-1) or (n+1) must be mutiples of 2, the other must be mutiple of 4.
so (n-1)(n+1) must be multiple of 8
when (n-1)(n+1) = 24, r =0
when (n-1)(n+1) = 32, r =8
Not Suff.

(1)
if 3 is not a factor of n, either (n-1) or (n+1) must be mutiples of 3
so (n-1)(n+1) must be multiple of 3
when (n-1)(n+1) = 24, r =0
when (n-1)(n+1) = 27, r =3
Not Suff

(1)&(2)
Since (n-1)(n+1) is a multiple of 8 and 3, it must be a multiple of 24.
So r = 0
Suff.

Ans is C

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This is not a quality discussion. It has been retired.

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Re: Remainder--DS   [#permalink] 18 Aug 2008, 09:00
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