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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]

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18 Sep 2011, 19:19
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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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19 Sep 2011, 03:15
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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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Kudos [?]: 2131 [4], given: 376

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19 Sep 2011, 22:22
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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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Karishma
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 18096 [3], given: 236 Math Expert Joined: 02 Sep 2009 Posts: 42544 Kudos [?]: 135217 [3], given: 12673 Re: What is the remainder when the positive integer n is divided [#permalink] Show Tags 04 Jul 2013, 03:12 3 This post received KUDOS Expert's post 3 This post was BOOKMARKED 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 _________________ Kudos [?]: 135217 [3], given: 12673 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7788 Kudos [?]: 18096 [3], given: 236 Location: Pune, India Re: What is the remainder when the positive integer n is divided [#permalink] Show Tags 04 Jul 2013, 20:07 3 This post received KUDOS Expert's post 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)) $$(K+1)^3 = K^3 + 3K^2 + 3K + 1$$ _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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18 Sep 2011, 19:41
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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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19 Sep 2011, 03:19
2
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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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19 Sep 2011, 03:59
1
KUDOS
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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19 Sep 2011, 03:27
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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19 Sep 2011, 03:30
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??

Kudos [?]: 184 [0], given: 42

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20 Sep 2011, 00:24
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]

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04 Jul 2013, 03:00
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] 04 Jul 2013, 03:00
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