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DS - remainder

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by pinktiger » Wed Jun 18, 2008 11:03 am
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.

Pls explain your approach.
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Source: — Data Sufficiency |
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by egybs » Wed Jun 18, 2008 11:43 am
This is a really cool question.

Answer is C.

Factors of 24 are 2,2,2,3

1) try out a couple odd numbers...
n=3 yields 6... R=6
n=5 yields 24... R =0
So since both are different, INSUFFICIENT

2) try out a couple non-multiple of three numbers:
n=5 yields 24... R=0
n=8 yields 63... R=15
So since both are different, INSUFFICIENT

Together:
n=5 yields 24... R=0
n=7 yields 48... R=0
n=11 yields 120... R=0
n=13 yields 168... R=0
.....
so, SUFFICIENT!

Why?

Since we're multiplying n-1 and n+1, we're multiplying the two adjacent numbers. Whenever n is odd, we'll be multiplying two even numbers together... one is at least divisible by 2^2 while the other is divisible by 2... So we have at least 2x2x2 as factors.. Recall that 2,2,2,3 are the factors of 24. When n is not divisible by 3, we also end up with either n+1 or n-1 being divisible by 3, since picking any 3 adjacent numbers will yield one number that is divisible by three. So under those circumstances, 24 will always be a factor... always leaving a remainder of 0.
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by pinktiger » Wed Jun 18, 2008 11:59 am
Thanks buddy!
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by Ian Stewart » Wed Jun 18, 2008 11:59 am
It's likely clear that neither statement is sufficient on its own.

Note that n-1, n, and n+1 are three consecutive numbers. One of them must be divisible by 3. From statement 2, n is not, so either n-1 or n+1 is.

-From statement 1, n is odd. Thus, n-1 and n+1 are consecutive even numbers. One must be a multiple of 2, the other a multiple of 4.

Thus (n-1)*(n+1) is divisible by 2*4*3 = 24. The remainder will be zero. C.

edit- was posting at the same time as egybs!
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