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If p is a positive integer, what is the greatest integer that must be

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If p is a positive integer, what is the greatest integer that must be  [#permalink]

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New post Updated on: 12 Dec 2018, 03:25
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A
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If p is a positive integer, what is the greatest integer that must be a factor of p^6 - p^2?


A. 5
B. 12
C. 19
D. 25
E. 16

Originally posted by ritu1009 on 12 Dec 2018, 03:17.
Last edited by ritu1009 on 12 Dec 2018, 03:25, edited 2 times in total.
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If p is a positive integer, what is the greatest integer that must be  [#permalink]

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New post Updated on: 15 Dec 2018, 10:26
I started plugging numbers:

if p = 1, the result would be zero, and any number can be claimed as a factor of zero.

if p = 2, 2^6 - 2^2 = 60, which eliminates options C,D,E instantly.
5 and 12 are factors of 60, but 12 is bigger.

so B.
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Originally posted by Mahmoudfawzy83 on 12 Dec 2018, 12:29.
Last edited by Mahmoudfawzy83 on 15 Dec 2018, 10:26, edited 1 time in total.
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Re: If p is a positive integer, what is the greatest integer that must be  [#permalink]

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New post 14 Dec 2018, 12:27
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Hi there!

Let me contribute with some observations in this matter:

01. It is easy to prove that the expression given may be factorized as p^2 (p^2+1) (p+1) (p-1) and from that, to conclude that this expression is always divisible by 12 (for p positive integer).

02. The fact is that p^6 - p^2 = p (p^5-p) and it can be proved that p^5 - p is always divisible by 5 (for p positive integer) (*), therefore the correct answer is 12*5 = 60, not among the alternative choices.

Mahmoudfawzy83 : 3^6 -3^2 is 720

Regards,
Fabio.

P.S.: (*) Just consider the factorization of (p^5 - p) and take into account that p must be equal to 5M, 5M+1, 5M+2, 5M+3 or 5M+4 , where M is integer. All cases are trivial.
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Re: If p is a positive integer, what is the greatest integer that must be  [#permalink]

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New post 15 Dec 2018, 10:27
fskilnik wrote:
Hi there!

Let me contribute with some observations in this matter:

01. It is easy to prove that the expression given may be factorized as p^2 (p^2+1) (p+1) (p-1) and from that, to conclude that this expression is always divisible by 12 (for p positive integer).

02. The fact is that p^6 - p^2 = p (p^5-p) and it can be proved that p^5 - p is always divisible by 5 (for p positive integer) (*), therefore the correct answer is 12*5 = 60, not among the alternative choices.

Mahmoudfawzy83 : 3^6 -3^2 is 720

Regards,
Fabio.

P.S.: (*) Just consider the factorization of (p^5 - p) and take into account that p must be equal to 5M, 5M+1, 5M+2, 5M+3 or 5M+4 , where M is integer. All cases are trivial.



I answered in right by mistake :lol:
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Re: If p is a positive integer, what is the greatest integer that must be  [#permalink]

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New post 16 Dec 2018, 04:15
Mahmoudfawzy83 wrote:

I answered in right by mistake :lol:

Hello again , Mahmoudfawzy83 !

Thanks for the kudos.

I would like to congratulate you for your approach, in my opinion the BEST way to manage GMAT Problem Solving problems like this one.

I explain:

You may (and SHOULD) suppose the problem is well-posed, in the sense that there is ONLY one CORRECT alternative choice among the five given.

In other words, your reasoning for refuting some alternative choices based on a particular case exploration is GREAT.

Among the "survivors", it is also commendable to FOCUS on the property asked (maximum value), but in this case just one small detail is important:

To be sure 12 "would be" the right answer, 12 must be a divisor of EVERY expression p^6-p^2, I mean, of p^6-p^2 for every value of positive integer p. That´s the case!

(The fact that 60 is not among the possibilities, is an examiner´s fault. It is his/her burden to guarantee that you, the test taker, is offered the right answer among the ones given!)

I hope you (and other readers) benefit from those details!

Regards and success in your studies,
Fabio.

P.S.: "any number can be claimed as a factor of zero." must be understood as "any integer different from zero can be claimed as a factor of zero".
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If p is a positive integer, what is the greatest integer that must be  [#permalink]

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New post 20 Dec 2018, 08:49
Hi,

Can you please explain solution "01"?

How were you able to determine that p^2 (p^2+1) (p+1) (p-1) will be divisble by 12?

i solved by expressing (p^2-1) (p^2) (p^2+1) where 25 can also fit in considering p may be 5.

Where am i wrong here.

Thanks

fskilnik wrote:
Hi there!

Let me contribute with some observations in this matter:

01. It is easy to prove that the expression given may be factorized as p^2 (p^2+1) (p+1) (p-1) and from that, to conclude that this expression is always divisible by 12 (for p positive integer).

02. The fact is that p^6 - p^2 = p (p^5-p) and it can be proved that p^5 - p is always divisible by 5 (for p positive integer) (*), therefore the correct answer is 12*5 = 60, not among the alternative choices.

Mahmoudfawzy83 : 3^6 -3^2 is 720

Regards,
Fabio.

P.S.: (*) Just consider the factorization of (p^5 - p) and take into account that p must be equal to 5M, 5M+1, 5M+2, 5M+3 or 5M+4 , where M is integer. All cases are trivial.
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If p is a positive integer, what is the greatest integer that must be  [#permalink]

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New post Updated on: 20 Dec 2018, 12:30
Sajjad1093 wrote:
Hi,

Can you please explain solution "01"?

How were you able to determine that p^2 (p^2+1) (p+1) (p-1) will be divisble by 12?

i solved by expressing (p^2-1) (p^2) (p^2+1) where 25 can also fit in considering p may be 5.

Where am i wrong here.

Thanks

fskilnik wrote:
Hi there!

Let me contribute with some observations in this matter:

01. It is easy to prove that the expression given may be factorized as p^2 (p^2+1) (p+1) (p-1) and from that, to conclude that this expression is always divisible by 12 (for p positive integer).

02. The fact is that p^6 - p^2 = p (p^5-p) and it can be proved that p^5 - p is always divisible by 5 (for p positive integer) (*), therefore the correct answer is 12*5 = 60, not among the alternative choices.

Mahmoudfawzy83 : 3^6 -3^2 is 720

Regards,
Fabio.

P.S.: (*) Just consider the factorization of (p^5 - p) and take into account that p must be equal to 5M, 5M+1, 5M+2, 5M+3 or 5M+4 , where M is integer. All cases are trivial.


Greetings Sajjad1093

for your second question about 25:
25 can fit if p = 5. that's correct , but can't always fit.
in the question: MUST is a keyword

I am tagging fskilnik as well so we can learn from him
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Originally posted by Mahmoudfawzy83 on 20 Dec 2018, 09:30.
Last edited by Mahmoudfawzy83 on 20 Dec 2018, 12:30, edited 1 time in total.
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If p is a positive integer, what is the greatest integer that must be  [#permalink]

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New post 20 Dec 2018, 11:52
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Hi Sajjad1093 and Mahmoudfawzy83 ,

It is a pleasure to help you both understand my reasoning in my statement (1):

01. It is easy to prove that the expression given may be factorized as p^2 (p^2+1) (p+1) (p-1) and from that, to conclude that this expression is always divisible by 12 (for p positive integer).

Consider ANY positive integer p fixed. For this value of p, we have:

\({p^6} - {p^2} = {p^2}\left( {{p^4} - 1} \right) = {p^2}\left( {{p^2} + 1} \right)\left( {{p^2} - 1} \right) = {p^2}\left( {{p^2} + 1} \right)\left( {p + 1} \right)\left( {p - 1} \right) = A\left( p \right) \cdot B\left( p \right)\)

\(A\left( p \right) = \left( {p - 1} \right)p\left( {p + 1} \right)\,\,{\rm{is}}\,\,{\rm{even}}\,\,\left( {{\rm{consecutive}}\,\,{\rm{integers}}} \right)\,\,{\rm{and}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,3\,\,\,\,\,\left( {3\,\,{\rm{consecutive}}\,\,{\rm{integers}}} \right)\,\,\,\,\, \Rightarrow \,\,\,\,A\left( p \right) = 6M,\,\,M\,\,{\mathop{\rm int}}\)

\(B\left( p \right) = p\left( {{p^2} + 1} \right)\,\,{\rm{is}}\,\,{\rm{even}}\,\,\,\,\left( {{\rm{if}}\,\,p\,\,{\rm{odd}}\,,\,\,{p^2}\,\,{\rm{is}}\,\,{\rm{odd}}\,\,{\rm{hence}}\,\,{p^2} + 1\,\,{\rm{is}}\,\,{\rm{even}}} \right)\,\,\,\, \Rightarrow \,\,\,\,B\left( p \right) = 2N,\,\,N\,\,{\mathop{\rm int}}\)

\({p^6} - {p^2} = 12MN\,\,,\,\,MN\,\,{\mathop{\rm int}} \,\,\,\,\, \Rightarrow \,\,\,\,{p^6} - {p^2}\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,12\,\,\,\,\)


Please tag me again, if needed.

Regards and success in your studies,
Fabio.

P.S.: if you want my explanation for my statement (2), tag me about it, too.
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Re: If p is a positive integer, what is the greatest integer that must be  [#permalink]

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New post 20 Dec 2018, 12:27
Thank fskilnik

please, can you explain how p^6 - P^4 is divisible by 5?
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Re: If p is a positive integer, what is the greatest integer that must be  [#permalink]

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New post 20 Dec 2018, 12:51
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Mahmoudfawzy83 wrote:
Thank fskilnik

please, can you explain how p^6 - p^2 is divisible by 5?

Sure, let´s prove my statement 02., repeated below for convenience:

02. The fact is that p^6 - p^2 = p (p^5-p) and it can be proved that p^5 - p is always divisible by 5 (for p positive integer).

Consider ANY positive integer p fixed.

We know (from the Division Algorithm, or simply "remainders understanding") that p must be of one (only) of the following possibilities:
p = 5M , 5M+1 , 5M+2 , 5M+3 or 5M+4 , where M is a (nonnegative) integer.

\({p^5} - p = p\left( {{p^4} - 1} \right) = p\left( {{p^2} + 1} \right)\left( {p + 1} \right)\left( {p - 1} \right)\)

\(p = 5M\,\,\,\, \Rightarrow \,\,\,\,\,{p^5} - p = 5M\left( {{p^2} + 1} \right)\left( {p + 1} \right)\left( {p - 1} \right) = 5M \cdot {\mathop{\rm int}} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{done}}\)

\(p = 5M + 1\,\,\,\, \Rightarrow \,\,\,\,\,{p^5} - p = p\left( {{p^2} + 1} \right)\left( {p + 1} \right)\left( {5M} \right) = 5M \cdot {\mathop{\rm int}} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{done}}\)

\(p = 5M + 2\,\,\,\, \Rightarrow \,\,\,\,\,{p^5} - p = p \cdot \underbrace {\left[ {25{M^2} + 20M + 4 + 1} \right]}_{5\left( {5{M^2} + 4M + 1} \right)\,\, = \,5N\,,\,\,N\,\,{\mathop{\rm int}} }\left( {p + 1} \right)\left( {p - 1} \right) = 5N \cdot {\mathop{\rm int}} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{done}}\)

\(p = 5M + 3\,\,\,\, \Rightarrow \,\,\,\,\,{p^5} - p = p \cdot \underbrace {\left[ {25{M^2} + 30M + 9 + 1} \right]}_{5\left( {5{M^2} + 6M + 2} \right)\,\, = \,5N\,,\,\,N\,\,{\mathop{\rm int}} }\left( {p + 1} \right)\left( {p - 1} \right) = 5N \cdot {\mathop{\rm int}} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{done}}\)

\(p = 5M + 4\,\,\,\, \Rightarrow \,\,\,\,\,{p^5} - p = p\left( {{p^2} + 1} \right)\underbrace {\left( {5M + 5} \right)}_{5\left( {M + 1} \right)\,\, = 5N\,\,,\,\,N\,\,{\mathop{\rm int}} }\left( {p - 1} \right) = 5N \cdot {\mathop{\rm int}} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{done}}\)

Regards,
Fabio.

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Re: If p is a positive integer, what is the greatest integer that must be  [#permalink]

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New post 20 Dec 2018, 14:20
Could a kind soul help me out here?

How I went about thinking about p^2(P^2)(P-1)(P+1) is that no matter what the largest factor would be a positive root +1.

I've read each post and can't get it to click. What do I not see?
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Re: If p is a positive integer, what is the greatest integer that must be  [#permalink]

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New post 21 Dec 2018, 09:45
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The expression p^6 - p^2 can be expressed as follow:
p^6 - p^2=(p^2 +1)*p^2*(p+1)*p*(p-1)

As we can see, (p+1)*p*(p-1) is a set of 3 consecutive number. Therefore, the product of these numbers is divisible for 1,2,3 (1)

Moreover, (p^2 +1)*p^2 is also a set of 2 consecutive number. Therefore, the product of these 2 numbers is divisible for 1 & 2 (2)

From (1) and (2), we can conclude that the product of these 5 numbers is divisible for 1,2,3,4,6 & 12.
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Re: If p is a positive integer, what is the greatest integer that must be &nbs [#permalink] 21 Dec 2018, 09:45
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