GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 18 Sep 2019, 13:23

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

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

Author Message
TAGS:

### Hide Tags

Intern
Joined: 20 Feb 2012
Posts: 28
If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

23 Feb 2012, 08:10
14
98
00:00

Difficulty:

85% (hard)

Question Stats:

54% (02:19) correct 46% (01:59) wrong based on 960 sessions

### HideShow timer Statistics

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.
Math Expert
Joined: 02 Sep 2009
Posts: 58092
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

23 Feb 2012, 08:20
44
54
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.

_________________
##### General Discussion
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9637
Location: Pune, India
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

14 Dec 2013, 01:37
2
4
audiogal101 wrote:

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/
_________________
Karishma
Veritas Prep GMAT Instructor

Math Expert
Joined: 02 Sep 2009
Posts: 58092
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

06 Nov 2012, 05:21
1
1
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.

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.
_________________
SVP
Joined: 06 Sep 2013
Posts: 1604
Concentration: Finance
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

27 Jan 2014, 18:18
1
VeritasPrepKarishma wrote:
audiogal101 wrote:

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
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4475
If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

12 May 2014, 16:27
1
BANON wrote:
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.

I'm happy to help with it here.

The first skill we need is the "rebuilding the dividend" equation. See:
http://magoosh.com/gmat/2012/gmat-quant ... emainders/
If we divide N by D, and get a quotient of Q with remainder of R, then
N/D = Q + (R/D)
or
N = Q*D + R
That is one of the most powerful and unappreciated formulas on the GMAT, the "rebuilding the dividend" equation. As for the name, the terminology here is
N = the dividend, the thing divided
D = the divisor, the number by which we are dividing
Q = the quotient, the result of division
R = the remainder

Here, we can change the statements into equations:
Statement #1: (p + n) = 5*S + 1, for some quotient integer S
Statement #2: (p - n) = 3*T + 1, for some quotient integer T

The second thing that you really need to recognize quickly on the GMAT is the Difference of Two Squares patterns, the single most important algebraic pattern on the entire GMAT. See:
http://magoosh.com/gmat/2012/gmat-quant ... o-squares/
http://magoosh.com/gmat/2013/three-alge ... -the-gmat/
Here, this implies that:
p^2 - n^2 = (p + n)*(p - n)

One statement talks only about 5, and the other, only about 3, so neither is sufficient individually. If both are considered together, we can multiply the two formulas together:
p^2 - n^2 = (p + n)*(p - n) = (5*S + 1)*(3*T + 1) = 15*ST + 5*S + 3*T + 1
Well, clearly the first term, (15*ST) would be divisible by 15. The problem is: we don't know the values of S or T. We know they are integers, but we don't know their values. If think about possible values of 5*S, multiples of 5, then when we divide multiples of 5 by 15, we could have remainders of 0, 5, or 10. Similarly, when we divided 3*T, multiples of 3, by 15, we could have remainders of 0, 3, 6, 9, or 12. Thus, 15 will go evenly into the first term, but may have a remainder on the second term and on the third time, and of course, will have a remainder of 1 on the last term. Because there's still choice about the possible remainders, even with both statements combined, we cannot answer the prompt question.

Even with both statements combined, everything is still insufficient. Answer = (E)

Does all this make sense?
Mike
_________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9637
Location: Pune, India
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

02 Nov 2018, 07:47
1
once we've narrowed down to C/E
can we test the relationship in a simultaneous equation setup by adding p+n = 5Q+1 and p-n = 3K+1 which
works out to 2p = 5Q + 3K + 2 i.e. three variables requiring three assumptions hence unsolvable ?
Thus E is the best answer.

No, this does not work out. There is no point looking for p. We need p^2 - n^2. Sometimes, you need less information to get the value of what you need even though you may not be able to get the value of each variable independently.

p + n = 5Q + 1

p - n = 3K + 1

Now, what you need is (p + n)*(p - n)

(p + n)*(p - n) = (5Q + 1)*(3K + 1) = 15QK + 3K + 5Q + 1

Is this divisible by 15? We know that the first term (15QK) is but what about the sum of the other 3 terms? We don't know since that depends on the values of Q and K.
If K = 5 and Q = 3, this is not divisible by 15.
If K = 3 and Q = 4, this is divisible by 15.
Hence, both statements are insufficient.
_________________
Karishma
Veritas Prep GMAT Instructor

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9637
Location: Pune, India
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

11 Dec 2018, 06:01
1
singh8891 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.

Hello
When the question stem says that The remainder when p + n is divided by 5 is 1, doesn't it implicitly place a constraint on p-n? So dont we need to check p-n too?
While solving the question, I calculated p+n based on the criteria, p-n for the chosen numbers and then p^2-n^2 and finally checked divisibility. I started running out of time and had to guess and move.

Note the information given to you about (p + n). The remainder when you divide it by 5 is 1.

So (p+n) can be 6/11/16/21/26/31/36... etc. This gives us limitless values for (p-n) with all possible different remainders when divided by 5.

Yes, a particular small value for (p+n) could have put a constraint on (p - n). Such as (p+n) = 6. Now (p - n) could take a few values only.
_________________
Karishma
Veritas Prep GMAT Instructor

Manager
Joined: 22 Dec 2011
Posts: 218
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

04 Nov 2012, 22:16
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.

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
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9637
Location: Pune, India
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

14 Dec 2013, 01:03
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.
_________________
Karishma
Veritas Prep GMAT Instructor

Intern
Joined: 23 Oct 2012
Posts: 23
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

14 Dec 2013, 01:18

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!?
Intern
Joined: 09 Oct 2014
Posts: 5
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

17 Oct 2014, 13:15
BANON wrote:
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.

i plugged in a 9 for p and a 2 for n.

i don't get the theory,,, so when i get confused i just try to solve it.
11/5 has a remainder of 1, i get other cans, so ns.

9-2 = 7= divided by 3 is a 1. granted i know others may have this.

but to have a remainder of one, and to be be dived by both a factor of 1 and 5, this fits. thus i plugged in the numbers for 7^2 = 49 - 2^2= 4 thus equals 45/15 = 0

i assumed that the remainder 15 was a factor of the 5 and 3. thus i picked c.

how in the world would i know not to do this?

thanks,.
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9637
Location: Pune, India
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

18 Oct 2014, 08:19
BANON wrote:
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.

i plugged in a 9 for p and a 2 for n.

i don't get the theory,,, so when i get confused i just try to solve it.
11/5 has a remainder of 1, i get other cans, so ns.

9-2 = 7= divided by 3 is a 1. granted i know others may have this.

but to have a remainder of one, and to be be dived by both a factor of 1 and 5, this fits. thus i plugged in the numbers for 7^2 = 49 - 2^2= 4 thus equals 45/15 = 0

i assumed that the remainder 15 was a factor of the 5 and 3. thus i picked c.

how in the world would i know not to do this?

thanks,.

How do you know that for every suitable value of p and n, the remainder will always be the same?

Say p = 5 and n = 1.
p+n = 6 when divided by 5 gives remainder 1.
p-n = 4 when divided by 3 gives remainder 1.

p^2 - n^2 = 25 - 1 = 24
When this is divided by 15, remainder is 9. But you got remainder 0 with your values. Hence the remainder can take different values.

You cannot be sure that every time the remainder obtained will be 0. You cannot use number plugging to prove something. You cannot be sure that you have tried every possible value which could give a different result. You will need to rely on theory to prove that the remainder obtained will be the same in each case (or may not be the same). Using number plugging to disprove something is possible since all you have to do is find that one value for which the result does not hold.
_________________
Karishma
Veritas Prep GMAT Instructor

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9637
Location: Pune, India
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

05 Jan 2015, 21:28
Quote:
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.

You have explained the above question in one of the way. But if we think in the below way can u point out what is the mistake I am making?

p+n = 5t +1 : 1,6, 11, 16,21, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76..........
p-n = 3m + 1 : 1, 4,7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79.......

Hence the general formula tht can be written is = 15 k + 1 & hence the remainder shall be 1 ??

Please let me where is the gap in the understanding??

Think on a few points:

The general formula 15k+1 is the formula for what? What number will be in this format?
Are you saying that the format of p^2 - n^2 will be this? If yes, then think - why?

p+n = 5t +1
p-n = 3m + 1

$$p^2 - n^2 = (p+n)(p-n) = (5t+1)(3m+1) = 15tm + 5t + 3m + 1$$

How did you get (15k+1)? How can we be sure that 5t+3m is divisible by 15?

The same thing was done by another reader above and I explained why this is wrong here: if-p-and-n-are-positive-integers-and-p-n-what-is-the-128002.html#p1304072
_________________
Karishma
Veritas Prep GMAT Instructor

Director
Joined: 07 Aug 2011
Posts: 505
GMAT 1: 630 Q49 V27
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

17 Mar 2015, 11:32
BANON wrote:
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.

Option A:
At P=5 N=1 P+N when divided by 5 will have remainder 1 AND $$P^2-N^2 / 15$$ will have remainder 9.
At P=15 N=1 P+N when divided by 5 will have remainder 1 AND $$P^2-N^2 / 15$$ will have remainder 14.

Option B:
At P=5 N=1 P-N when divided by 3 will have remainder 1 AND $$P^2-N^2 / 15$$ will have remainder 9.
At P=6 N=2 P-N when divided by 3 will have remainder 1 AND $$P^2-N^2 / 15$$ will have remainder 2.

Both A and B are not sufficient.

Combining
P+N = 5K + 1 i.e. and ODD Integer
P-N = 3X+1 i.e. an EVEN Integer and so it has no say on the divisibility by 15 it acts as a constant in P^2-N^2.
(P+N) will decided whether $$P^2 - N^2$$ is divisible by 15 and also the remainder , but we have already proved in option A that P+N can have two different remainders (9,14 , we can test more numbers) and also (P-N) when multiplied to these two remainders will definitely give different remainders when divided by 15.

Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 7885
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

17 Feb 2018, 23:09
BANON wrote:
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.

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 2 variables (p and n) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first.

Conditions 1) & 2):
$$p + n = 6, p - n = 4 : p = 5, n = 1 : p^2 - n^2 = 25 - 1 = 24$$, its remainder is $$4$$.
$$p + n = 11, p - n = 7 : p = 9, n = 2 : p^2 - n^2 = 81 - 4 = 77$$, its remainder is $$2$$.

Thus, both conditions together are not sufficient, since the remainder is not unique.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only \$79 for 1 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"
Intern
Joined: 14 Jun 2018
Posts: 5
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

01 Nov 2018, 22:55
once we've narrowed down to C/E
can we test the relationship in a simultaneous equation setup by adding p+n = 5Q+1 and p-n = 3K+1 which
works out to 2p = 5Q + 3K + 2 i.e. three variables requiring three assumptions hence unsolvable ?
Thus E is the best answer.
Manager
Joined: 01 May 2018
Posts: 64
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

09 Dec 2018, 07:07
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.

Hello
When the question stem says that The remainder when p + n is divided by 5 is 1, doesn't it implicitly place a constraint on p-n? So dont we need to check p-n too?
While solving the question, I calculated p+n based on the criteria, p-n for the chosen numbers and then p^2-n^2 and finally checked divisibility. I started running out of time and had to guess and move.
Director
Joined: 24 Oct 2016
Posts: 511
GMAT 1: 670 Q46 V36
GMAT 2: 690 Q47 V38
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

03 Aug 2019, 12:40
BANON wrote:
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.

p^2 - n^2 = (p + n)(p - n)

(p + n)(p - n) = 15a + r
r = ?

1) (p + n) = 5a + 1 => Not sufficient
2) (p - n) = 3b + 1 => Not sufficient
1 + 2)
(5a + 1) (3b + 1) = 15ab + 5a + 3b + 1

Don't know whether 5a + 3b + 1 is a multiple of 15 => Not sufficient => E
_________________

If you found my post useful, KUDOS are much appreciated. Giving Kudos is a great way to thank and motivate contributors, without costing you anything.
Intern
Joined: 17 Jun 2019
Posts: 19
Location: Israel
GPA: 3.95
WE: Engineering (Computer Hardware)
Re: If p and n are positive integers and p > n, what is the remainder when  [#permalink]

### Show Tags

07 Sep 2019, 03:13
wasn't paying attention, stupid mistake
Re: If p and n are positive integers and p > n, what is the remainder when   [#permalink] 07 Sep 2019, 03:13
Display posts from previous: Sort by