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If x and y are positive integers greater than 3, what is the remainder

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If x and y are positive integers greater than 3, what is the remainder [#permalink]

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GMAT Club's Fresh Challenge Problem.



If x and y are positive integers greater than 3, what is the remainder when xy is divided by 9?

(1) The positive difference between x and y is 2.
(2) x and y are both primes numbers.
[Reveal] Spoiler: OA

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Re: If x and y are positive integers greater than 3, what is the remainder [#permalink]

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New post 06 Dec 2017, 10:30
Imo C

St1:
X=9, y=7

Remainder = 0

X=6, y= 4
Remainder = 6

Not sufficient

St2:
X=5
Y=7
Remainder = 8

X=5
Y= 5
Remainder = 7
Not sufficient

Combining 1 and 2

X=9, y=7
X=13, y=11
X= 19, y= 17

All gives same remainder 8

Sufficient



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Re: If x and y are positive integers greater than 3, what is the remainder [#permalink]

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New post 06 Dec 2017, 10:52
Bunuel wrote:

GMAT Club's Fresh Challenge Problem.



If x and y are positive integers greater than 3, what is the remainder when xy is divided by 9?

(1) The positive difference between x and y is 2.
(2) x and y are both primes numbers.



a proper method would be

(1) The positive difference between x and y is 2.
many cases possible..
9 and 7, remainder is 0
7 and 5 , remainder is 8
insuff

(2) x and y are both primes numbers.
5 and 7 will give 8 as remainder
7 and 11 will give 5 as remainder
Insuff

Combined
the numbers would be of type 6n+1 and 6n-1..
product = \((6n+1)(6n-1) = 36n^2-1\)..
\(36n^2\) is div by 9, so remainder will be -1
but remainder has to be positive so 9-1=8
suff

C
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html


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Re: If x and y are positive integers greater than 3, what is the remainder [#permalink]

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New post 09 Dec 2017, 01:27
chetan2u wrote:
Bunuel wrote:

GMAT Club's Fresh Challenge Problem.



If x and y are positive integers greater than 3, what is the remainder when xy is divided by 9?

(1) The positive difference between x and y is 2.
(2) x and y are both primes numbers.



a proper method would be

(1) The positive difference between x and y is 2.
many cases possible..
9 and 7, remainder is 0
7 and 5 , remainder is 8
insuff

(2) x and y are both primes numbers.
5 and 7 will give 8 as remainder
7 and 11 will give 5 as remainder
Insuff

Combined
the numbers would be of type 6n+1 and 6n-1..
product = \((6n+1)(6n-1) = 36n^2-1\)..
\(36n^2\) is div by 9, so remainder will be -1
but remainder has to be positive so 9-1=8
suff

C

How did you get numbers of that form 6n+1 and 6n-1?
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Re: If x and y are positive integers greater than 3, what is the remainder [#permalink]

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New post 09 Dec 2017, 04:33
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stickman wrote:
chetan2u wrote:
Bunuel wrote:

GMAT Club's Fresh Challenge Problem.



If x and y are positive integers greater than 3, what is the remainder when xy is divided by 9?

(1) The positive difference between x and y is 2.
(2) x and y are both primes numbers.



a proper method would be

(1) The positive difference between x and y is 2.
many cases possible..
9 and 7, remainder is 0
7 and 5 , remainder is 8
insuff

(2) x and y are both primes numbers.
5 and 7 will give 8 as remainder
7 and 11 will give 5 as remainder
Insuff

Combined
the numbers would be of type 6n+1 and 6n-1..
product = \((6n+1)(6n-1) = 36n^2-1\)..
\(36n^2\) is div by 9, so remainder will be -1
but remainder has to be positive so 9-1=8
suff

C

How did you get numbers of that form 6n+1 and 6n-1?


All the PRIMES above 3 will be ODD and NOT divisible by 2 and 3
so 6n+1 and 6n-1 will give you all numbers not div by 2 and 3
AND only two consecutive odd numbers will be prime as the THIRD will be surely be multiple of 3..
example 5,7,9........9,11,13.....11,13,15
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html


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Re: If x and y are positive integers greater than 3, what is the remainder [#permalink]

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New post 09 Dec 2017, 04:43
chetan2u wrote:
stickman wrote:
How did you get numbers of that form 6n+1 and 6n-1?


All the PRIMES above 3 will be ODD and NOT divisible by 2 and 3
so 6n+1 and 6n-1 will give you all numbers not div by 2 and 3
AND only two consecutive odd numbers will be prime as the THIRD will be surely be multiple of 3..
example 5,7,9........9,11,13.....11,13,15


Perfect thanks.

So say the same qs was rephrased but instead of greater than 3, it was greater than 5, would it be 30n-1 and 30n+1?
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If x and y are positive integers greater than 3, what is the remainder [#permalink]

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New post 09 Dec 2017, 04:57
stickman wrote:
chetan2u wrote:
stickman wrote:
How did you get numbers of that form 6n+1 and 6n-1?


All the PRIMES above 3 will be ODD and NOT divisible by 2 and 3
so 6n+1 and 6n-1 will give you all numbers not div by 2 and 3
AND only two consecutive odd numbers will be prime as the THIRD will be surely be multiple of 3..
example 5,7,9........9,11,13.....11,13,15


Perfect thanks.

So say the same qs was rephrased but instead of greater than 3, it was greater than 5, would it be 30n-1 and 30n+1?
..


No, it will always remain 6n+1 and 6n-1..

30n+1 and 30n-1 will miss many prime numbers ..
if n =1 , the number becomes 31 and 29, both are prime but it misses out on many other prime like 7,11,13..

reason we take 6n+1 and 6n-1 is that MOST of the numbers are div by 2 and 3 and therefore, possibility of missing any prime is 0..
n=1.....7 and 5
n=2.....13 and 11
n=3.....19 and 17

so here we are avoiding EVEN numbers and multiples of 3 - 9,15,21 and so on..

BUT remember it is not necessary that 6n+1 and 6n-1 will be prime but PRIME will always be 6n+1 and 6n-1
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html


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If x and y are positive integers greater than 3, what is the remainder [#permalink]

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New post 09 Dec 2017, 05:00
chetan2u wrote:

No, it will always remain 6n+1 and 6n-1..

30n+1 and 30n-1 will miss many prime numbers ..
if n =1 , the number becomes 31 and 29, both are prime but it misses out on many other prime like 7,11,13..

reason we take 6n+1 and 6n-1 is that MOST of the numbers are div by 2 and 3 and therefore, possibility of missing any prime is 0..
n=1.....7 and 5
n=2.....13 and 11
n=3.....19 and 17

so here we are avoiding EVEN numbers and multiples of 3 - 9,15,21 and so on..

BUT remember it is not necessary that 6n+1 and 6n-1 will be prime but PRIME will always be 6n+1 and 6n-1


I understand - thank you.
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Re: If x and y are positive integers greater than 3, what is the remainder [#permalink]

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New post 09 Dec 2017, 05:07
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stickman wrote:
chetan2u wrote:

No, it will always remain 6n+1 and 6n-1..

30n+1 and 30n-1 will miss many prime numbers ..
if n =1 , the number becomes 31 and 29, both are prime but it misses out on many other prime like 7,11,13..

reason we take 6n+1 and 6n-1 is that MOST of the numbers are div by 2 and 3 and therefore, possibility of missing any prime is 0..
n=1.....7 and 5
n=2.....13 and 11
n=3.....19 and 17

so here we are avoiding EVEN numbers and multiples of 3 - 9,15,21 and so on..

BUT remember it is not necessary that 6n+1 and 6n-1 will be prime but PRIME will always be 6n+1 and 6n-1


I understand - thank you.


chetan2u is referring to the following property:

Any prime number p, which is greater than 3, could be expressed as \(p=6n+1\) or \(p=6n+5\) or \(p=6n-1\), where n is an integer greater than 1.

Any prime number p, which is greater than 3, when divided by 6 can only give the remainder of 1 or 5 (remainder cannot be 2 or 4 as in this case p would be even and the remainder cannot be 3 as in this case p would be divisible by 3).

So, any prime number p, which is greater than 3, could be expressed as \(p=6n+1\) or \(p=6n+5\) or \(p=6n-1\), where n is an integer greater than 1.

But:
Not all number which yield a remainder of 1 or 5 upon division by 6 are primes, so vise-versa of the above property is not true. For example 25 yields the remainder of 1 upon division be 6 and it's not a prime number.

Hope it's clear.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
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. ,11 Mixed Questions, 12 Fresh Meat

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., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Re: If x and y are positive integers greater than 3, what is the remainder   [#permalink] 09 Dec 2017, 05:07
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