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What is the remainder when the positive integer n is divided

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What is the remainder when the positive integer n is divided [#permalink] New post 18 Sep 2011, 18:19
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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

OPEN DISCUSSION OF THIS QUESTION IS HERE: what-is-the-remainder-when-the-positive-integer-n-is-divided-96366.html
[Reveal] Spoiler: OA
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Re: Number propierties [#permalink] New post 18 Sep 2011, 18:41
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andresfigue wrote:
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


Algebraic way:

1) eXPAND (K+1)^3 = k^3+3.(K^2)+3k+1. So definitely remainder of 1 as the first 3 terms are multiples of K. Sufficient

2) Insufficient.

Numerical way:

Take k= 2 & 3 . u will get remainder 1 in both cases.

So A.
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Re: Number propierties [#permalink] New post 19 Sep 2011, 02:15
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andresfigue wrote:
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


Sol:

1)
\frac{(k+1)^3}{k}

Remainder:
\frac{(k+1)^3}{k}=Remainder Of(\frac{Remainder Of(\frac{k+1}{k})*Remainder Of(\frac{k+1}{k})*Remainder Of(\frac{k+1}{k})}{k})=Remainder Of(\frac{1*1*1}{k})=1

Sufficient.

Ans: "A"
************************************************

I think there is some principle of induction that we can apply here.

For more on the formula I used to solve this:
compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html
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Re: Number propierties [#permalink] New post 19 Sep 2011, 02:19
fluke wrote:
andresfigue wrote:
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


Sol:

1)
\frac{(k+1)^3}{k}

Remainder:
\frac{(k+1)^3}{k}=Remainder Of(\frac{Remainder Of(\frac{k+1}{k})*Remainder Of(\frac{k+1}{k})*Remainder Of(\frac{k+1}{k})}{k})=Remainder Of(\frac{1*1*1}{k})=1

Sufficient.

Ans: "A"
************************************************

I think there is some principle of induction that we can apply here.

For more on the formula I used to solve this:
compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html


Gud Explanation Fluke. Hope my explanations above were also correct although a little traditional :)
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Re: Number propierties [#permalink] New post 19 Sep 2011, 02:27
GMATPASSION wrote:
Gud Explanation Fluke. Hope my explanations above were also correct although a little traditional :)


Oh, absolutely!! In fact, Kudos for that.

I believe you used the concept of mathematical induction, in which all terms but one are divisible by the denominator. I remember Karishma's describing it once. I don't remember that exactly.
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Re: Number propierties [#permalink] New post 19 Sep 2011, 02:30
fluke wrote:
GMATPASSION wrote:
Gud Explanation Fluke. Hope my explanations above were also correct although a little traditional :)


Oh, absolutely!! In fact, Kudos for that.

I believe you used the concept of mathematical induction, in which all terms but one are divisible by the denominator. I remember Karishma's describing it once. I don't remember that exactly.


Thanks for my first kudos buddy. 'Mathematical Induction' Wats dat? Never Heard of that??
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Re: Number propierties [#permalink] New post 19 Sep 2011, 02:59
GMATPASSION wrote:
andresfigue wrote:
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


Algebraic way:

1) eXPAND (K+1)^3 = k^3+3.(K^2)+3k+1. So definitely remainder of 1 as the first 3 terms are multiples of K. Sufficient

2) Insufficient.

Numerical way:

Take k= 2 & 3 . u will get remainder 1 in both cases.

So A.


Good explanation GMATPASSION. Kudos for that :)
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Re: Number propierties [#permalink] New post 19 Sep 2011, 21:22
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GMATPASSION wrote:

Thanks for my first kudos buddy. 'Mathematical Induction' Wats dat? Never Heard of that??


Responding to a PM:

Actually it was a discussion on 'Binomial Theorem' (Induction is an altogether different concept which is out of GMAT scope)
Binomial theorem comes in handy in many remainder questions.

With a power of 3, it is easy to expand the expression and see that only 1 will be the remainder (as GMATPASSION did). For higher powers, binomial theorem can be used. I have put up a post on the Veritas blog discussing it and its applications. Here is the link. Get back in case there are any doubts.

http://www.veritasprep.com/blog/2011/05 ... ek-in-you/
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Re: Number propierties [#permalink] New post 19 Sep 2011, 23:24
VeritasPrepKarishma wrote:
GMATPASSION wrote:

Thanks for my first kudos buddy. 'Mathematical Induction' Wats dat? Never Heard of that??


Responding to a PM:

Actually it was a discussion on 'Binomial Theorem' (Induction is an altogether different concept which is out of GMAT scope)
Binomial theorem comes in handy in many remainder questions.

With a power of 3, it is easy to expand the expression and see that only 1 will be the remainder (as GMATPASSION did). For higher powers, binomial theorem can be used. I have put up a post on the Veritas blog discussing it and its applications. Here is the link. Get back in case there are any doubts.

http://www.veritasprep.com/blog/2011/05 ... ek-in-you/


Got it!!! thanks a lot Karishma.
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Re: What is the remainder when the positive integer n is divided [#permalink] New post 04 Jul 2013, 02:00
The expansion of (K+1)^3 needs to be memorized? BTW great explanation @fluke
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Re: What is the remainder when the positive integer n is divided [#permalink] New post 04 Jul 2013, 02:12
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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= k^3 + 3k^2 + 3k + 1=k(k^2+3k+3)+1 --> first term, k(k^2+3k+3), is obviously divisible by k and 1 divide by k yields the remainder of 1 (as k>1). Sufficient.

(2) k=5. Know nothing about n, hence insufficient.

Answer: A.

OPEN DISCUSSION OF THIS QUESTION IS HERE: what-is-the-remainder-when-the-positive-integer-n-is-divided-96366.html
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Re: What is the remainder when the positive integer n is divided [#permalink] New post 04 Jul 2013, 19:07
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fozzzy wrote:
The expansion of (K+1)^3 needs to be memorized? BTW great explanation @fluke


In case you do forget the expansion/don't know it, just multiply:

(K+1)^3 = (K+1)(K^2 + 2K + 1) (We certainly know the expansion of (K+1)^2 or we can find it my multiplying (K+1)(K+1))
(K+1)^3 = K^3 + 3K^2 + 3K + 1
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Re: What is the remainder when the positive integer n is divided   [#permalink] 04 Jul 2013, 19:07
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