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If p and n are positive integers and p > n, what is the [#permalink]
 23 Feb 2012, 08:10
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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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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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If p and n are positive integers and p > n, what is the [#permalink]
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Had the question been for divisibility rather than for remainder, the answer wud had been C.
But since it asks for remainder, the answer is E.
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If p and n are positive integers and p > n [#permalink]
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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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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: If p and n are positive integers and p > n, what is the [#permalink]
07 Jun 2012, 16:02
Hey Kyro, could you please explain how we can apply the factoring in case the querstion asks for divisibility?
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Re: If p and n are positive integers and p > n, what is the [#permalink]
07 Jun 2012, 16:03
Or anyone* I don't know if Kyro is still on this forum..
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If P and N are positive integers and P is greater than N, [#permalink]
16 Aug 2012, 12:33
If P and N are positive integers and P is greater than N, what is the reminder when P^2 - N^2 is divided by 15?
I) The reminder when P+N is divided by 5 is 1.
II) the reminder when P-N is divided by 3 is 1.
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Re: If P and N are positive integers and P is greater than N, [#permalink]
16 Aug 2012, 12:37
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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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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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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. NEW!!!
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. NEW!!!
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