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Is p^2 - 1 divisible by 12?

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Is p^2 - 1 divisible by 12?  [#permalink]

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Updated on: 15 Aug 2014, 09:13
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Question Stats:

67% (01:30) correct 33% (01:23) wrong based on 195 sessions

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Is p^2 - 1 divisible by 12?

(1) p > 3

(2) p is a prime number

Source: 4gmat

Originally posted by alphonsa on 15 Aug 2014, 08:13.
Last edited by Bunuel on 15 Aug 2014, 09:13, edited 1 time in total.
Edited the question
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Re: Is p^2 - 1 divisible by 12?  [#permalink]

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15 Aug 2014, 09:23
2
4
Is p^2 - 1 divisible by 12?

(1) p > 3. This one is clearly insufficient: if p = 4, then the answer is NO but if p = 5, then the answer is YES. Not sufficient.

(2) p is a prime number. If p = 2, then the answer is NO but if p = 5, then the answer is YES. Not sufficient.

(1)+(2) Important property:
ANY prime number $$p$$ greater than 3 can 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.

So, according to the above, p can be expressed as $$p=6n+1$$ or $$p=6n-1$$. If $$p=6n+1$$, then $$p^2 - 1 = 36n^2 +12n=12(3n^2+1)$$ and if $$p=6n-1$$, then $$p^2 - 1 = 36n^2 -12n=12(3n^2-1)$$. In both cases p is a multiple of 12. Sufficient.

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Re: Is p^2 - 1 divisible by 12?  [#permalink]

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16 Aug 2014, 19:27
Hi Bunuel,
for P = 5 or 7 , p^2-1 is divisible by 12.

Bunuel wrote:
Is p^2 - 1 divisible by 12?

(1) p > 3. This one is clearly insufficient: if p = 4, then the answer is NO but if p = 5, then the answer is YES. Not sufficient.

(2) p is a prime number. If p = 2, then the answer is NO but if p = 5, then the answer is YES. Not sufficient.

(1)+(2) Important property:
ANY prime number $$p$$ greater than 3 can 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.

So, according to the above, p can be expressed as $$p=6n+1$$ or $$p=6n-1$$. If $$p=6n+1$$, then $$p^2 - 1 = 36n^2 +12n=12(3n^2+1)$$ and if $$p=6n-1$$, then $$p^2 - 1 = 36n^2 -12n=12(3n^2-1)$$. In both cases p is a multiple of 12. Sufficient.

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Re: Is p^2 - 1 divisible by 12?  [#permalink]

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17 Aug 2014, 04:05
sunita123 wrote:
Hi Bunuel,
for P = 5 or 7 , p^2-1 is divisible by 12.

Bunuel wrote:
Is p^2 - 1 divisible by 12?

(1) p > 3. This one is clearly insufficient: if p = 4, then the answer is NO but if p = 5, then the answer is YES. Not sufficient.

(2) p is a prime number. If p = 2, then the answer is NO but if p = 5, then the answer is YES. Not sufficient.

(1)+(2) Important property:
ANY prime number $$p$$ greater than 3 can 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.

So, according to the above, p can be expressed as $$p=6n+1$$ or $$p=6n-1$$. If $$p=6n+1$$, then $$p^2 - 1 = 36n^2 +12n=12(3n^2+1)$$ and if $$p=6n-1$$, then $$p^2 - 1 = 36n^2 -12n=12(3n^2-1)$$. In both cases p is a multiple of 12. Sufficient.

For any prime number p greater than 3, p^2 - 1 IS divisible by 12. So, taken together the statements are sufficient to get a definite YES answer to the question. Which means that the answer is C.
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Re: Is p^2 - 1 divisible by 12?  [#permalink]

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17 Aug 2014, 16:24
ANY prime number $$p$$ greater than 3 can 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.

Again, I have never known the concept before. Thank you, Bunuel
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Is p^2 - 1 divisible by 12?  [#permalink]

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05 Oct 2014, 12:17
ANY prime number p greater than 3 can be expressed as p=6n+1 or p=6n+5 (p=6n-1), where n is an integer >1.

I have a problem with highlighted part. How can we represent 5 or 7 in the forms 6n +1 or 6n +5. I think the above property holds good for n>=0. Can somebody clarify?
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Re: Is p^2 - 1 divisible by 12?  [#permalink]

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06 Oct 2014, 00:31
annie2014 wrote:
ANY prime number p greater than 3 can be expressed as p=6n+1 or p=6n+5 (p=6n-1), where n is an integer >1.

I have a problem with highlighted part. How can we represent 5 or 7 in the forms 6n +1 or 6n +5. I think the above property holds good for n>=0. Can somebody clarify?

It should be >=1.

5 = 6*1 - 1;
7 = 6*1 + 1.
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Re: Is p^2 - 1 divisible by 12?  [#permalink]

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08 Oct 2014, 09:44
Thank you Bunnel for your prompt response! You are truly incredible!
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Is p^2 - 1 divisible by 12?  [#permalink]

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05 Jan 2019, 09:04
alphonsa wrote:
Is p^2 - 1 divisible by 12?

(1) p > 3

(2) p is a prime number

Source: 4gmat

$$p^2 - 1$$
can be written as (p-1) (p+1)

Statement 1 -> p=4,5,6
Case 1: 4, $$\frac{5*6}{12}$$ , Question will be answered by a No

Case 2 : 5, $$\frac{4*6}{12}$$, Question will be answered by a Yes

Statement 2-> p =2,5
Case 1: 2, $$\frac{1*3}{12}$$, Question will be answered by a No

Case 2 : 5, $$\frac{4*6}{12}$$, Question will be answered by a Yes

Combine, you will get a Yes if you take the case as p = 5, 7
Case 1 : 5, $$\frac{4*6}{12}$$

Case 2: 7, $$\frac{8*6}{12}$$

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Is p^2 - 1 divisible by 12?   [#permalink] 05 Jan 2019, 09:04
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