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# What is the remainder when the positive integer n is divided

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Manager
Joined: 05 Jan 2009
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What is the remainder when the positive integer n is divided [#permalink]  01 Jul 2009, 10:40
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What is the remainder when the positive integer n is divided by the positive integer k,
where k > 1?
(1) n = (k+1)^3
(2) k = 5
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
Senior Manager
Joined: 04 Jun 2008
Posts: 305
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Kudos [?]: 112 [0], given: 15

Re: What is the remainder ? [#permalink]  01 Jul 2009, 10:45
Ans should be A

You can pick numbers for K and N that satisfy sttmt 1 and check, everytime remainder will come as 1.

You can also use a principle that says -

if N/k gives remainder = x

then aN/k will give remainder ax (as long as ax < k)

if N was k+1, remainder would be 1 always
but if N = (k+1)^3, remainder will be 1^3

B is not sufficient.
Manager
Joined: 05 Jan 2009
Posts: 83
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Kudos [?]: 24 [0], given: 2

Re: What is the remainder ? [#permalink]  01 Jul 2009, 12:28
if N = (k+1)^3, remainder will be 1^3--can you please explain this stm in detail?
Senior Manager
Joined: 23 Jun 2009
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Location: Turkey
Schools: UPenn, UMich, HKS, UCB, Chicago
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Re: What is the remainder ? [#permalink]  01 Jul 2009, 12:38
(k+1)^3=k^3+3k^2+3k+1. So the remainder is always 1
Senior Manager
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Re: What is the remainder ? [#permalink]  01 Jul 2009, 12:44
There is a remainder theorem that states

If x/y gives remainder z

then ax/y will give remainder az, as long as az is < y

if p/y gives remainder q

then p*x/y will give remainder z*q........ as long as zq is < y

N = (k+1)^3

K+1/k will always give remainder of 1

so when (k+1)(k+1)(K+1) is divided by K, i will give remainder of 1*1*1.....ie 1

But we have to be carefull while using this theorem, its safe as long as remainder is 1, otherwise care has to be taken that the remainder doesn't exceed the divisor, because in that case, the rule cannot be applied.
Manager
Joined: 16 Apr 2009
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Schools: Ross
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Re: What is the remainder ? [#permalink]  01 Jul 2009, 12:48
Quote:
What is the remainder when the positive integer n is divided by the positive integer k,
where k > 1?
(1) n = (k+1)^3
(2) k = 5

Stat 1

plug in numbers

n = (k+1)^3

Let K be 2
n=(2+1)^3
n=3^3 =9
n/k=9/2 ---the remainder is 1

n = (k+1)^3

Let K be 3
n=(3+1)^3
n=4^3 =64
n/k=64/3 ---the remainder is 1

n = (k+1)^3

Let K be 4
n=(4+1)^3
n=5^3 =125
n/k=125/4 ---the remainder is 1

A is suff

stat 2.
k = 5
we don't know about n..hence not suff.
_________________

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Manager
Joined: 16 Apr 2009
Posts: 246
Schools: Ross
Followers: 2

Kudos [?]: 38 [0], given: 10

Re: What is the remainder ? [#permalink]  01 Jul 2009, 12:55
Quote:
(k+1)^3=k^3+3k^2+3k+1. So the remainder is always 1

I liked the tip.
Thanks!
_________________

Keep trying no matter how hard it seems, it will get easier.

Re: What is the remainder ?   [#permalink] 01 Jul 2009, 12:55
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