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If p and n are positive integers and p > n, what is the

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If p and n are positive integers and p > n, what is the [#permalink] New post 23 Feb 2012, 07:10
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If p and n are positive integers and p > n, what is the remainder when p^2 - n^2 is divided by 15 ?

(1) The remainder when p + n is divided by 5 is 1.
(2) The remainder when p - n is divided by 3 is 1.
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Re: DATA5 [#permalink] New post 23 Feb 2012, 07:20
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If p and n are positive integers and p>n, what is the remainder when p^2 - n^2 is divided by 15?

First of all p^2 - n^2=(p+n)(p-n).

(1) The remainder when p + n is divided by 5 is 1. No info about p-n. Not sufficient.

(2) The remainder when p - n is divided by 3 is 1. No info about p+n. Not sufficient.

(1)+(2) "The remainder when p + n is divided by 5 is 1" can be expressed as p+n=5t+1 and "The remainder when p - n is divided by 3 is 1" can be expressed as p-n=3k+1.

Multiply these two --> (p+n)(p-n)=(5t+1)(3k+1)=15kt+5t+3k+1, now first term (15kt) is clearly divisible by 15 (r=0), but we don't know about 5t+3k+1. For example t=1 and k=1, answer r=9 BUT t=7 and k=3, answer r=0. Not sufficient.

OR by number plugging: if p+n=11 (11 divided by 5 yields remainder of 1) and p-n=1 (1 divided by 3 yields remainder of 1) then (p+n)(p-n)=11 and remainder upon division 11 by 15 is 11 BUT if p+n=21 (21 divided by 5 yields remainder of 1) and p-n=1 (1 divided by 3 yields remainder of 1) then (p+n)(p-n)=21 and remainder upon division 21 by 15 is 6. Not sufficient.

Answer: E.
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Re: DATA5 [#permalink] New post 05 Nov 2012, 03:33
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Jp27 wrote:
Bunuel wrote:
If p and n are positive integers and p>n, what is the remainder when p^2 - n^2 is divided by 15?

First of all p^2 - n^2=(p+n)(p-n).

(1) The remainder when p + n is divided by 5 is 1. No info about p-n. Not sufficient.

(2) The remainder when p - n is divided by 3 is 1. No info about p+n. Not sufficient.

(1)+(2) "The remainder when p + n is divided by 5 is 1" can be expressed as p+n=5t+1 and "The remainder when p - n is divided by 3 is 1" can be expressed as p-n=3k+1.

Multiply these two --> (p+n)(p-n)=(5t+1)(3k+1)=15kt+5t+3k+1, now first term (15kt) is clearly divisible by 15 (r=0), but we don't know about 5t+3k+1. For example t=1 and k=1, answer r=9 BUT t=7 and k=3, answer r=0. Not sufficient.

OR by number plugging: if p+n=11 (11 divided by 5 yields remainder of 1) and p-n=1 (1 divided by 3 yields remainder of 1) then (p+n)(p-n)=11 and remainder upon division 11 by 15 is 11 BUT if p+n=21 (21 divided by 5 yields remainder of 1) and p-n=1 (1 divided by 3 yields remainder of 1) then (p+n)(p-n)=21 and remainder upon division 21 by 15 is 6. Not sufficient.

Answer: E.


Hi Bunuel - 1 doubt.. why can't the below process be followed?

p+n = 5A+1 => 1,6,11,16,21,26
p-n = 3B+1 => 1,4,7,10,13,16,19,21

p+n * p-n => 15 K + 16. Hence the remainder on division by 15 gives 1.

Cheers

try picking numbers:

if p= 5 and n = 1 p+n = 6 remainder after dividing by 5 is 1
p-n= 4 remainder after dividing by 3 is 1

p^2 -N2 = 25-1 = 24 remainder after dividing by 15 is 9

now consider p =9 and n =2

p+n = 11 remainder after dividing by 5 is 1
p-n = 7 remainder after dividing by 3 is 1

p^2 - N^2 = 81-4 = 77 remainder after dividing by 15 is 2.

hence both statement combined are not suff
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Re: Remainder question [#permalink] New post 14 Nov 2013, 15:06
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TiagoMagalhaes wrote:
Dear all,
Can anyone please provide me the solution for the following problem?
If p and n are positive integers and p > n, what is the remainder when p^2 - n^2 is divided by 15?
(1) The remainder when p + n is divided by 5 is 1
(2) The remainder when p - n is divided by 3 is 1

Thanks in advance.
Regards,
Tiago Magalhães


Merging similar topics. Please refer to the solutions above.
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Re: DATA5 [#permalink] New post 14 Dec 2013, 00:37
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audiogal101 wrote:
Thanks for your response Karishma.

I can see how based on your examples that my interpretation is not correct. However, i was going based off of a little tip in one of the other posts...

3) If a number leaves a remainder ‘r’ (the number is the divisor), all its factors will have the same remainder ‘r’ provided the value of ‘r’ is less than the value of the factor.
Eg. If remainder of a number when divided by 21 is 5, then the remainder of that same number when divided by 7 (which is a factor of 21) will also be 5.

How is the situation in the problem different ...wherein the statement is not holding true!?


That is fine.

But this does not imply that if one factor leaves a remainder 'r', all multiples will leave remainder 'r' too. On the other hand, if all factors of that particular multiple leave the remainder 'r', then the multiple will leave remainder 'r' too.

Take an example:
If n is divided by 21 and it leaves remainder 1, when n is divided by 7, remainder will still be 1. When n is divided by 3, remainder will still be 1. - Correct

If n is divided by 7 and it leaves remainder 1, it doesn't mean that when n is divided by 14/21/28/35 ... it will leave remainder 1 in all cases.

But if n divided by 7 leaves remainder 1 and when divided by 3 leaves a remainder 1 too, it will leave remainder 1 when divided by 21 too.

For an explanation of these, check:
http://www.veritasprep.com/blog/2011/04 ... unraveled/
http://www.veritasprep.com/blog/2011/04 ... y-applied/
http://www.veritasprep.com/blog/2011/05 ... emainders/
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Re: DATA5 [#permalink] New post 04 Nov 2012, 21:16
Bunuel wrote:
If p and n are positive integers and p>n, what is the remainder when p^2 - n^2 is divided by 15?

First of all p^2 - n^2=(p+n)(p-n).

(1) The remainder when p + n is divided by 5 is 1. No info about p-n. Not sufficient.

(2) The remainder when p - n is divided by 3 is 1. No info about p+n. Not sufficient.

(1)+(2) "The remainder when p + n is divided by 5 is 1" can be expressed as p+n=5t+1 and "The remainder when p - n is divided by 3 is 1" can be expressed as p-n=3k+1.

Multiply these two --> (p+n)(p-n)=(5t+1)(3k+1)=15kt+5t+3k+1, now first term (15kt) is clearly divisible by 15 (r=0), but we don't know about 5t+3k+1. For example t=1 and k=1, answer r=9 BUT t=7 and k=3, answer r=0. Not sufficient.

OR by number plugging: if p+n=11 (11 divided by 5 yields remainder of 1) and p-n=1 (1 divided by 3 yields remainder of 1) then (p+n)(p-n)=11 and remainder upon division 11 by 15 is 11 BUT if p+n=21 (21 divided by 5 yields remainder of 1) and p-n=1 (1 divided by 3 yields remainder of 1) then (p+n)(p-n)=21 and remainder upon division 21 by 15 is 6. Not sufficient.

Answer: E.


Hi Bunuel - 1 doubt.. why can't the below process be followed?

p+n = 5A+1 => 1,6,11,16,21,26
p-n = 3B+1 => 1,4,7,10,13,16,19,21

p+n * p-n => 15 K + 16. Hence the remainder on division by 15 gives 1.

Cheers
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Re: DATA5 [#permalink] New post 06 Nov 2012, 04:21
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Jp27 wrote:
Bunuel wrote:
If p and n are positive integers and p>n, what is the remainder when p^2 - n^2 is divided by 15?

First of all p^2 - n^2=(p+n)(p-n).

(1) The remainder when p + n is divided by 5 is 1. No info about p-n. Not sufficient.

(2) The remainder when p - n is divided by 3 is 1. No info about p+n. Not sufficient.

(1)+(2) "The remainder when p + n is divided by 5 is 1" can be expressed as p+n=5t+1 and "The remainder when p - n is divided by 3 is 1" can be expressed as p-n=3k+1.

Multiply these two --> (p+n)(p-n)=(5t+1)(3k+1)=15kt+5t+3k+1, now first term (15kt) is clearly divisible by 15 (r=0), but we don't know about 5t+3k+1. For example t=1 and k=1, answer r=9 BUT t=7 and k=3, answer r=0. Not sufficient.

OR by number plugging: if p+n=11 (11 divided by 5 yields remainder of 1) and p-n=1 (1 divided by 3 yields remainder of 1) then (p+n)(p-n)=11 and remainder upon division 11 by 15 is 11 BUT if p+n=21 (21 divided by 5 yields remainder of 1) and p-n=1 (1 divided by 3 yields remainder of 1) then (p+n)(p-n)=21 and remainder upon division 21 by 15 is 6. Not sufficient.

Answer: E.


Hi Bunuel - 1 doubt.. why can't the below process be followed?

p+n = 5A+1 => 1,6,11,16,21,26
p-n = 3B+1 => 1,4,7,10,13,16,19,21

p+n * p-n => 15 K + 16. Hence the remainder on division by 15 gives 1.

Cheers


p+n = 5A+1 and p-n = 3B+1 does not mean that (p+n)*(p-n)=15K+16. When you expand (p+n)*(p-n)=(5A+1)(3B+1) you won't get an expression of the form 15K+16.
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Remainder question [#permalink] New post 14 Nov 2013, 15:04
Dear all,
Can anyone please provide me the solution for the following problem?
If p and n are positive integers and p > n, what is the remainder when p^2 - n^2 is divided by 15?
(1) The remainder when p + n is divided by 5 is 1
(2) The remainder when p - n is divided by 3 is 1

Thanks in advance.
Regards,
Tiago Magalhães
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Re: DATA5 [#permalink] New post 13 Dec 2013, 01:34
Bunuel wrote:
If p and n are positive integers and p>n, what is the remainder when p^2 - n^2 is divided by 15?

First of all p^2 - n^2=(p+n)(p-n).

(1) The remainder when p + n is divided by 5 is 1. No info about p-n. Not sufficient.

(2) The remainder when p - n is divided by 3 is 1. No info about p+n. Not sufficient.

(1)+(2) "The remainder when p + n is divided by 5 is 1" can be expressed as p+n=5t+1 and "The remainder when p - n is divided by 3 is 1" can be expressed as p-n=3k+1.

Multiply these two --> (p+n)(p-n)=(5t+1)(3k+1)=15kt+5t+3k+1, now first term (15kt) is clearly divisible by 15 (r=0), but we don't know about 5t+3k+1. For example t=1 and k=1, answer r=9 BUT t=7 and k=3, answer r=0. Not sufficient.

OR by number plugging: if p+n=11 (11 divided by 5 yields remainder of 1) and p-n=1 (1 divided by 3 yields remainder of 1) then (p+n)(p-n)=11 and remainder upon division 11 by 15 is 11 BUT if p+n=21 (21 divided by 5 yields remainder of 1) and p-n=1 (1 divided by 3 yields remainder of 1) then (p+n)(p-n)=21 and remainder upon division 21 by 15 is 6. Not sufficient.

Answer: E.



Hi Bunuel,

I reasoned this question as follows:

Prompt is asking what is remainder when... (p+n) (p-n) /15 ?

St 1> gives remainder when p+n is divided by 5 is 1...this implies..remainder when (p+n) is divided by 15 is also 1...(since 5 is a factor of 15). ..NOT SUFF
St 2> gives remainder when p-n is divided by 3 is 1...this implies..remainder when (p-n) is divided by 15 is also 1...(since 3 is a factor of 15) ..not SUFF

Combinign 1 & 2... (p+n) /15 gives remainder 1, and (p-n) /15 gives remainder 1.....so (p+n) (p-n)/15 ...should also yield remainder 1...since we can multiply the remainders in this case...because the divisor is the same. . Please clarify, is this correct?? If not , why?
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Re: DATA5 [#permalink] New post 14 Dec 2013, 00:03
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audiogal101 wrote:

Prompt is asking what is remainder when... (p+n) (p-n) /15 ?

St 1> gives remainder when p+n is divided by 5 is 1...this implies..remainder when (p+n) is divided by 15 is also 1...(since 5 is a factor of 15). ..NOT SUFF
St 2> gives remainder when p-n is divided by 3 is 1...this implies..remainder when (p-n) is divided by 15 is also 1...(since 3 is a factor of 15) ..not SUFF

Combinign 1 & 2... (p+n) /15 gives remainder 1, and (p-n) /15 gives remainder 1.....so (p+n) (p-n)/15 ...should also yield remainder 1...since we can multiply the remainders in this case...because the divisor is the same. . Please clarify, is this correct?? If not , why?


Responding to a pm:

The highlighted portion is incorrect.

Say p + n = 21.
When it is divided by 5, the remainder is 1. But when it is divided by 15, the remainder is 6. So your implication is not correct. You cannot say that the remainder when you divide p+n by 15 will also be 1.
All you can say is (p+n) = 5a + 1
Similarly, all you can say about statement 2 is (p-n) = 3b + 1

So using both also, you get (5a + 1)(3b + 1) = 15ab + 5a + 3b + 1
We know nothing about (5a + 3b + 1) - whether it is divisible by 15 or not.
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Re: DATA5 [#permalink] New post 14 Dec 2013, 00:18
Thanks for your response Karishma.

I can see how based on your examples that my interpretation is not correct. However, i was going based off of a little tip in one of the other posts...

3) If a number leaves a remainder ‘r’ (the number is the divisor), all its factors will have the same remainder ‘r’ provided the value of ‘r’ is less than the value of the factor.
Eg. If remainder of a number when divided by 21 is 5, then the remainder of that same number when divided by 7 (which is a factor of 21) will also be 5.

How is the situation in the problem different ...wherein the statement is not holding true!?
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Re: DATA5 [#permalink] New post 14 Dec 2013, 00:26
audiogal101 wrote:
Thanks for your response Karishma.

I can see how based on your examples that my interpretation is not correct. However, i was going based off of a little tip in one of the other posts...

3) If a number leaves a remainder ‘r’ (the number is the divisor), all its factors will have the same remainder ‘r’ provided the value of ‘r’ is less than the value of the factor.
Eg. If remainder of a number when divided by 21 is 5, then the remainder of that same number when divided by 7 (which is a factor of 21) will also be 5.

How is the situation in the problem different ...wherein the statement is not holding true!?



Ummm...i guess the statement is true...its just that 15 is not a factor of 5. Its the other way round.

So if the statement was (p+n) when divided by 15 gives a remainder of 1, then we can say (p+n) divided by 5 also gives a remainder of 1...
but if (p+n) leaves a remainder of 1 when divided by 5 , does not mean (p+n) divided by 15 will also leave remainder of 1...(as clearly evidenced by Karishma's example)
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Re: DATA5 [#permalink] New post 14 Dec 2013, 01:01
VeritasPrepKarishma wrote:
audiogal101 wrote:
Thanks for your response Karishma.

I can see how based on your examples that my interpretation is not correct. However, i was going based off of a little tip in one of the other posts...

3) If a number leaves a remainder ‘r’ (the number is the divisor), all its factors will have the same remainder ‘r’ provided the value of ‘r’ is less than the value of the factor.
Eg. If remainder of a number when divided by 21 is 5, then the remainder of that same number when divided by 7 (which is a factor of 21) will also be 5.

How is the situation in the problem different ...wherein the statement is not holding true!?


That is fine.

But this does not imply that if one factor leaves a remainder 'r', all multiples will leave remainder 'r' too. On the other hand, if all factors of that particular multiple leave the remainder 'r', then the multiple will leave remainder 'r' too.

Take an example:
If n is divided by 21 and it leaves remainder 1, when n is divided by 7, remainder will still be 1. When n is divided by 3, remainder will still be 1. - Correct

If n is divided by 7 and it leaves remainder 1, it doesn't mean that when n is divided by 14/21/28/35 ... it will leave remainder 1 in all cases.

But if n divided by 7 leaves remainder 1 and when divided by 3 leaves a remainder 1 too, it will leave remainder 1 when divided by 21 too.

For an explanation of these, check:
http://www.veritasprep.com/blog/2011/04 ... unraveled/
http://www.veritasprep.com/blog/2011/04 ... y-applied/
http://www.veritasprep.com/blog/2011/05 ... emainders/


Thanks a ton!! You do an excellent job of deconstructing a complex issue and presenting it in simple and interesting ways...
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Re: DATA5 [#permalink] New post 27 Jan 2014, 17:18
VeritasPrepKarishma wrote:
audiogal101 wrote:
Thanks for your response Karishma.

I can see how based on your examples that my interpretation is not correct. However, i was going based off of a little tip in one of the other posts...

3) If a number leaves a remainder ‘r’ (the number is the divisor), all its factors will have the same remainder ‘r’ provided the value of ‘r’ is less than the value of the factor.
Eg. If remainder of a number when divided by 21 is 5, then the remainder of that same number when divided by 7 (which is a factor of 21) will also be 5.

How is the situation in the problem different ...wherein the statement is not holding true!?


That is fine.

But this does not imply that if one factor leaves a remainder 'r', all multiples will leave remainder 'r' too. On the other hand, if all factors of that particular multiple leave the remainder 'r', then the multiple will leave remainder 'r' too.

Take an example:
If n is divided by 21 and it leaves remainder 1, when n is divided by 7, remainder will still be 1. When n is divided by 3, remainder will still be 1. - Correct

If n is divided by 7 and it leaves remainder 1, it doesn't mean that when n is divided by 14/21/28/35 ... it will leave remainder 1 in all cases.

But if n divided by 7 leaves remainder 1 and when divided by 3 leaves a remainder 1 too, it will leave remainder 1 when divided by 21 too.

For an explanation of these, check:
http://www.veritasprep.com/blog/2011/04 ... unraveled/
http://www.veritasprep.com/blog/2011/04 ... y-applied/
http://www.veritasprep.com/blog/2011/05 ... emainders/


I personally found this a bit tricky

For instance:

RoF x*y / n = Rof (x/n) * Rof (y/n) / n

Now when we have 15 and we can divide it into 5*3. I'm trying to figure out what's wrong with this approach cause I did the following

Remainder of p+n/5 as per the first statement is 1

Remainder of p-n/3 as per the second statement is 1

Therefore, remainder of product of 1*1 / 15 is 1

I thought this was C

Any clues?
Cheers!
J :)
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Re: DATA5 [#permalink] New post 27 Jan 2014, 20:33
Expert's post
jlgdr wrote:
I personally found this a bit tricky

For instance:

RoF x*y / n = Rof (x/n) * Rof (y/n) / n

Now when we have 15 and we can divide it into 5*3. I'm trying to figure out what's wrong with this approach cause I did the following

Remainder of p+n/5 as per the first statement is 1

Remainder of p-n/3 as per the second statement is 1

Therefore, remainder of product of 1*1 / 15 is 1

I thought this was C

Any clues?
Cheers!
J :)


This would have been correct had the situation been:

"Remainder of(p+n)/15 as per the first statement is 1
Remainder of (p-n)/15 as per the second statement is 1
Therefore, remainder of product of 1*1 / 15 is 1"

But the statements tell us what happens when (p+n) or (p-n) is divided by 5 or 3 not 15.

Statement 1: p+n = 5a + 1
Statement 2: p-n = 3b + 1

(p+n)(p-n) = 15ab + 5a + 3b + 1
Only the first term is divisible by 15. We don't know the remainder when the rest of the 3 terms are divided by 15.


On the other hand, had the statements been a little different, your method would have been correct.
Statement 1: p+n = 15a + 1
Statement 2: p-n = 15b + 1
(p+n)(p-n) = 15*15ab + 15a + 15b + 1
Now the remainder would be 1.
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Re: DATA5   [#permalink] 27 Jan 2014, 20:33
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