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If n is a prime number greater than 3, what is the remainder

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If n is a prime number greater than 3, what is the remainder [#permalink] New post 28 Oct 2011, 04:13
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If n is a prime number greater than 3, what is the remainder when n^2 is divided by 12?

A. 0
B. 1
C. 2
D. 3
E. 5

it's a simple quesiton, but the solutuion is inspiring.

[Reveal] Spoiler: Solution
n^2-1=(n-1)(n+1)
since (n-1) and (n+1) are consecutive even numbers,one of them can be divided by 2, another one can be divided by 4;
and because n can not be divided by 3, so one of (n-1) and (n+1) can be divided by 3.
So (n-1)(n+1)=n^2-1 is divisible by 24, then the remainder of n^2 divided by 24 is 1.
[Reveal] Spoiler: OA

Last edited by Bunuel on 24 Apr 2012, 11:46, edited 2 times in total.
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Re: A question with inspiring solution [#permalink] New post 28 Oct 2011, 05:25
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chonepiece wrote:
If n is a prime number greater than 3, what is the remainder when n is divided by 12?
0
1
2
3
5


This question is wrong. Please correct it.

what is the remainder when n is divided by 12
should be
what is the remainder when n^2 is divided by 12
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Re: If n is a prime number greater than 3, what is the remainder [#permalink] New post 09 Feb 2012, 13:57
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chonepiece wrote:
If n is a prime number greater than 3, what is the remainder when n^2 is divided by 12?
A. 0
B. 1
C. 2
D. 3
E. 5

it's a simple quesiton, but the solutuion is inspiring.

[Reveal] Spoiler: Solution
n^2-1=(n-1)(n+1)
since (n-1) and (n+1) are consecutive even numbers,one of them can be divided by 2, another one can be divided by 4;
and because n can not be divided by 3, so one of (n-1) and (n+1) can be divided by 3.
So (n-1)(n+1)=n^2-1 is divisible by 24, then the remainder of n^2 divided by 24 is 1.


There are several algebraic ways to solve this question including the one under the spoiler. But the easiest way is as follows: since we cannot have two correct answers just pick a prime greater than 3, square it and see what would be the remainder upon division of it by 12.

n=5 --> n^2=25 --> remainder upon division 25 by 12 is 1.

Answer: B.
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Re: If n is a prime number greater than 3, what is the remainder [#permalink] New post 10 Feb 2012, 01:33
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chonepiece wrote:
If n is a prime number greater than 3, what is the remainder when n^2 is divided by 12?
A. 0
B. 1
C. 2
D. 3
E. 5

it's a simple quesiton, but the solutuion is inspiring.

[Reveal] Spoiler: Solution
n^2-1=(n-1)(n+1)
since (n-1) and (n+1) are consecutive even numbers,one of them can be divided by 2, another one can be divided by 4;
and because n can not be divided by 3, so one of (n-1) and (n+1) can be divided by 3.
So (n-1)(n+1)=n^2-1 is divisible by 24, then the remainder of n^2 divided by 24 is 1.


Check out these posts for more on divisibility of consecutive integers:
http://www.veritasprep.com/blog/2011/09 ... c-or-math/
http://www.veritasprep.com/blog/2011/09 ... h-part-ii/

and of course, the most efficient solution would be what Bunuel suggested - Pick a prime number > 3 and check for it!
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Re: If n is a prime number greater than 3, what is the remainder [#permalink] New post 10 Feb 2012, 01:47
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all i did was pick numbers and try...got remainder as 1 always. so B. anything wrong in my approach pls advice
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Re: If n is a prime number greater than 3, what is the remainder [#permalink] New post 10 Feb 2012, 01:54
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sdas wrote:
all i did was pick numbers and try...got remainder as 1 always. so B. anything wrong in my approach pls advice


No. Nothing wrong. Just that you don't need to try many numbers. There will only be one answer to a PS question. So all you need to do is try any one number greater than 3. Whatever you get, that will be the answer in every case.
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Re: If n is a prime number greater than 3, what is the remainder [#permalink] New post 10 Feb 2012, 02:41
cool karishma. i only tried 5^2 and 7^2 both i got 1 as remainders
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Number properties [#permalink] New post 23 Apr 2012, 05:45
Hi

This is my first post, and am hoping someone can help me out with this question.

If n is a prime number greater than 3, what is the remainder when n^2 is divided by 12

A) 0
B) 1
C) 2
D) 3
E) 5

What I am looking for is advice on how to approach this problem, what are the math rules I can apply.

Many thanks
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Re: If n is a prime number greater than 3, what is the remainder [#permalink] New post 24 Apr 2012, 09:36
I tried solving it may be a lengthier method .. prime number greater than 6 is 6k+1 or 6k-1.. thus squaring say 6k+1 will give.. 36k2 + 12K + 1.... remainder 1... to double check for 5... 25/12..remainder 1... thus B.
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Re: If n is a prime number greater than 3, what is the remainder [#permalink] New post 07 May 2012, 16:02
Bunuel wrote:
chonepiece wrote:
If n is a prime number greater than 3, what is the remainder when n^2 is divided by 12?
A. 0
B. 1
C. 2
D. 3
E. 5

it's a simple quesiton, but the solutuion is inspiring.

[Reveal] Spoiler: Solution
n^2-1=(n-1)(n+1)
since (n-1) and (n+1) are consecutive even numbers,one of them can be divided by 2, another one can be divided by 4;
and because n can not be divided by 3, so one of (n-1) and (n+1) can be divided by 3.
So (n-1)(n+1)=n^2-1 is divisible by 24, then the remainder of n^2 divided by 24 is 1.


There are several algebraic ways to solve this question including the one under the spoiler. But the easiest way is as follows: since we can not have two correct answers just pick a prime greater than 3, square it and see what would be the remainder upon division of it by 12.

n=5 --> n^2=25 --> remainder upon division 25 by 12 is 1.

Answer: B.

Hello
if n is a prime number bigger than 3 it can also be 7 right ?
Hence 7 ^2 =49/12 Is not equal to 1
can I pick 7 or ?

thanks

best regards
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Re: If n is a prime number greater than 3, what is the remainder [#permalink] New post 08 May 2012, 00:18
Expert's post
keiraria wrote:
Bunuel wrote:
chonepiece wrote:
If n is a prime number greater than 3, what is the remainder when n^2 is divided by 12?
A. 0
B. 1
C. 2
D. 3
E. 5

it's a simple quesiton, but the solutuion is inspiring.

[Reveal] Spoiler: Solution
n^2-1=(n-1)(n+1)
since (n-1) and (n+1) are consecutive even numbers,one of them can be divided by 2, another one can be divided by 4;
and because n can not be divided by 3, so one of (n-1) and (n+1) can be divided by 3.
So (n-1)(n+1)=n^2-1 is divisible by 24, then the remainder of n^2 divided by 24 is 1.


There are several algebraic ways to solve this question including the one under the spoiler. But the easiest way is as follows: since we can not have two correct answers just pick a prime greater than 3, square it and see what would be the remainder upon division of it by 12.

n=5 --> n^2=25 --> remainder upon division 25 by 12 is 1.

Answer: B.

Hello
if n is a prime number bigger than 3 it can also be 7 right ?
Hence 7 ^2 =49/12 Is not equal to 1
can I pick 7 or ?

thanks

best regards


As explained. you can pick ANY prime greater than 3.

Also, 49 divided by 12 yields the remainder of 1: 49=4*12+1.

Hope it's clear.
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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: If n is a prime number greater than 3, what is the remainder [#permalink] New post 01 Nov 2012, 02:13
VeritasPrepKarishma wrote:
sdas wrote:
all i did was pick numbers and try...got remainder as 1 always. so B. anything wrong in my approach pls advice


No. Nothing wrong. Just that you don't need to try many numbers. There will only be one answer to a PS question. So all you need to do is try any one number greater than 3. Whatever you get, that will be the answer in every case.


I wonder why every prime no greater than 3 when squared and divided by 4 results in the remainder of 1 :shock:
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Re: If n is a prime number greater than 3, what is the remainder [#permalink] New post 01 Nov 2012, 04:54
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sachindia wrote:
VeritasPrepKarishma wrote:
sdas wrote:
all i did was pick numbers and try...got remainder as 1 always. so B. anything wrong in my approach pls advice


No. Nothing wrong. Just that you don't need to try many numbers. There will only be one answer to a PS question. So all you need to do is try any one number greater than 3. Whatever you get, that will be the answer in every case.


I wonder why every prime no greater than 3 when squared and divided by 4 results in the remainder of 1 :shock:


Any prime number p greater than 3 could be expressed as p=6n+1 or p=6n+5 (p=6n-1), where n is an integer >1.

That's because any prime number p greater than 3 when divided by 6 can only give remainder of 1 or 5 (remainder can not be 2 or 4 as in this case p would be even and remainder can not be 3 as in this case p would be divisible by 3).

But:
Note that, not all number which yield a remainder of 1 or 5 upon division by 6 are primes, so vise-versa of above property is not correct. For example 25 (for n=4) yields a remainder of 1 upon division by 6 and it's not a prime number.

Now, if a prime is of the form p=6n+1, then p^2=36n^2+12n+1=12(3n^2+n)+1 and if a prime is of the form p=6n-1, then p^2=36n^2-12n+1=12(3n^2-n)+1. Both yield the remainder of 1 when divided by 12.

Hope it's clear.
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COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: If n is a prime number greater than 3, what is the remainder [#permalink] New post 02 Nov 2012, 00:11
Do we really need to know this concept for GMAT :shock:
I find this going slighty above my head. . :roll:
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Re: If n is a prime number greater than 3, what is the remainder [#permalink] New post 02 Nov 2012, 01:14
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Re: If n is a prime number greater than 3, what is the remainder [#permalink] New post 17 Nov 2013, 12:39
If n is a prime number greater than 3, what is the remainder when n^2 is divided by 12?

A. 0
B. 1
C. 2
D. 3
E. 5

whenever any square of prime number>2 is divisible by 12 the remainder is always 1.
Check through options:

here in case n = 5,7,11 and so on.

5^2 = 25 when its divisible by 12,the remainder is 1.
7^2 = 49 when its divisible by 12, still the remainder is 1.
11^2 = 121 when its divisible by 12,the remaider is 1.

So answer of this question is (b).
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Re: If n is a prime number greater than 3, what is the remainder [#permalink] New post 01 May 2014, 02:13
pavanpuneet wrote:
I tried solving it may be a lengthier method .. prime number greater than 6 is 6k+1 or 6k-1.. thus squaring say 6k+1 will give.. 36k2 + 12K + 1.... remainder 1... to double check for 5... 25/12..remainder 1... thus B.



Too much of an overkill. As Bunuel suggested , here the best approach would be to put values and get the response.
Re: If n is a prime number greater than 3, what is the remainder   [#permalink] 01 May 2014, 02:13
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