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# If p is a positive integer, what is the remainder when p^2 – 1

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GMATWhiz Representative
Joined: 07 May 2019
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If p is a positive integer, what is the remainder when p^2 – 1  [#permalink]

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Updated on: 07 Apr 2020, 23:11
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39% (01:40) correct 61% (02:17) wrong based on 61 sessions

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If p is a positive integer, what is the remainder when $$p^2 – 1$$ is divided by 8?

(1) p is completely divisible by 3
(2) p leaves a remainder when divided by 2.

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Originally posted by GMATWhizTeam on 05 Apr 2020, 08:39.
Last edited by GMATWhizTeam on 07 Apr 2020, 23:11, edited 1 time in total.
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Re: If p is a positive integer, what is the remainder when p^2 – 1  [#permalink]

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05 Apr 2020, 08:59
2
p= pos int
(p^2-1)/8 ; remainder=?
(p-1)(p+1)/8
If p= odd, p-1 and p+1 both are even
And since p-1,p,p+1 are consecutive integers, one of p-1 and p+1 must be divisible by 2 and the other must be divisible by 4.
Keeping this is mind lets move to the statements
Statement 1: p is divisible by 3
If p=3, (p-1)(p+1)=8
So remainder=0
But if p=6,(p-1)(p+1)=35
Remainder=3
Insufficient
Statement 2: p is odd
Aa discussed above if p is odd, (p-1)(p+1) will always be divisible by 8
Sufficient
B

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Re: If p is a positive integer, what is the remainder when p^2 – 1  [#permalink]

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05 Apr 2020, 09:43
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If p is a positive integer, what is the remainder when p^2–1 is divided by 8?

(1) p is completely divisible by 3
if p=3, remainder is 0 BUT if p= 6, remainder is 3 . insufficient
(2) p leaves a remainder when divided by 2.
so, p = 2k+1 (k ≥0)
now, (p^2–1)/8 =((2k+1)^2-1 )/8 = (4k^2+4K) /8 = 4k(k+1)/8
k(k+1) always divisible by 2 since for any value of k one of these two , one must be divisible by 2
so, 4k(k+1) is divisible by 8 .
so, remainder = 0 . sufficient

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Re: If p is a positive integer, what is the remainder when p^2 – 1  [#permalink]

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05 Apr 2020, 16:23
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If p is a positive integer, what is the remainder when $$p^2 – 1$$ is divided by 8?

(1) p is completely divisible by 3
(2) p leaves a remainder when divided by 2.

Question stem analysis :-

is (p+1)(p-1) is divisible by 8?

if p = odd => (Odd+1)(Odd-1) = Even*Even = Even

Minimum value of p as odd = 1 => (p+1)(p-1) = 0 => divisible by 8
Next values of p as odd = 3 => (3+1)(3-1) = 4*2 => divisible by 8

if p = even => (Even+1)(Even-1) = Odd*Odd = Odd => Never divisible by 8

Question :- is p odd or even?

i) p = 3m => p = 3 => $$(p^2-1)$$ divisible by 8; p=12 => $$(p^2-1)$$ - Not divisible by 8 - Insufficient

ii) p leaves a remainder when divided by 2 => Remainder = 1 (As remainder can be only 0 or 1) => p is odd - Sufficient per the question stem analysis

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Re: If p is a positive integer, what is the remainder when p^2 – 1  [#permalink]

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05 Apr 2020, 21:26
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I am happy to see that all of you marked the answer correctly. Good job!

I hope you were able to internalize the first skill that we discussed in the Prepathon. Here's the detailed solution to the first question. Even if you got the question right make sure that you check out the method in the solution to ensure that you have solved it using the right method. All the best!

Feel free to post any doubts that you may have in the forum below. We will be happy to help you with your queries
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Score Improvement Strategy: How to score Q50+ on GMAT | 5 steps to Improve your Verbal score
Study Plan Articles: 3 mistakes to avoid while preparing | How to create a study plan? | The Right Order of Learning | Importance of Error Log
Helpful Quant Strategies: Filling Spaces Method | Avoid Double Counting in P&C | The Art of Not Assuming anything in DS | Number Line Method
Key Verbal Techniques: Plan-Goal Framework in CR | Quantifiers the tiny Game-changers | Countable vs Uncountable Nouns | Tackling Confusing Words in Main Point
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Re: If p is a positive integer, what is the remainder when p^2 – 1  [#permalink]

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06 Apr 2020, 06:15
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Hi sauravleo123, preetamsaha, shameekv1989 - You have marked the correct answers. Great job! We have also awarded you some points for the explanations that you provided in the comments. You can view your points on the Leaderboard section of the Megathread of the Prepathon.

You stand a good chance of winning prizes worth up to \$250 by participating in the Prepathon. Let us know if you would like to be tagged in the posts related to the Prepathon, we will be happy to tag you in these posts
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Re: If p is a positive integer, what is the remainder when p^2 – 1  [#permalink]

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06 Apr 2020, 06:22
GMATWhizTeam thanks for tagging me. btw the page is not opening.
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Re: If p is a positive integer, what is the remainder when p^2 – 1  [#permalink]

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06 Apr 2020, 06:26
preetamsaha wrote:
GMATWhizTeam thanks for tagging me. btw the page is not opening.

preetamsaha It's our pleasure. We will tag you in the other posts of the Prepathon. The link seems to be working for me. Try using this one. If this one also doesn't work you can find the Megathread of the Prepathon on the "GMAT Quantitative" main forum itself. Hope it helps
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Re: If p is a positive integer, what is the remainder when p^2 – 1  [#permalink]

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09 Apr 2020, 02:05
GMATWhizTeam wrote:
I am happy to see that all of you marked the answer correctly. Good job!

I hope you were able to internalize the first skill that we discussed in the Prepathon. Here's the detailed solution to the first question. Even if you got the question right make sure that you check out the method in the solution to ensure that you have solved it using the right method. All the best!

Feel free to post any doubts that you may have in the forum below. We will be happy to help you with your queries

Excellent video. Thanks a lot for your help
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Re: If p is a positive integer, what is the remainder when p^2 – 1   [#permalink] 09 Apr 2020, 02:05