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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]  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
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Re: Number propierties [#permalink]  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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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Manager Status: Retaking next month Affiliations: None Joined: 05 Mar 2011 Posts: 229 Location: India Concentration: Marketing, Entrepreneurship GMAT 1: 570 Q42 V27 GPA: 3.01 WE: Sales (Manufacturing) Followers: 5 Kudos [?]: 41 [2] , given: 42 Re: Number propierties [#permalink] 18 Sep 2011, 18:41 2 This post received KUDOS 1 This post was BOOKMARKED 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. Math Forum Moderator Joined: 20 Dec 2010 Posts: 2026 Followers: 145 Kudos [?]: 1259 [2] , given: 376 Re: Number propierties [#permalink] 19 Sep 2011, 02:15 2 This post received KUDOS 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 _________________ Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 5972 Location: Pune, India Followers: 1528 Kudos [?]: 8442 [2] , given: 194 Re: What is the remainder when the positive integer n is divided [#permalink] 04 Jul 2013, 19:07 2 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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Re: What is the remainder when the positive integer n is divided [#permalink]  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.

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: Number propierties [#permalink]  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]  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]  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]  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]  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]  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] 04 Jul 2013, 02:00
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