It is currently 23 Jan 2018, 10:12

### 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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

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

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 43381
If x and y are positive integers greater than 3, what is the remainder [#permalink]

### Show Tags

06 Dec 2017, 08:05
Expert's post
3
This post was
BOOKMARKED
00:00

Difficulty:

85% (hard)

Question Stats:

38% (01:20) correct 62% (01:31) wrong based on 54 sessions

### HideShow timer Statistics

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

_________________
Intern
Joined: 26 Oct 2017
Posts: 28
Re: If x and y are positive integers greater than 3, what is the remainder [#permalink]

### Show Tags

06 Dec 2017, 09:30
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

Sent from my ONEPLUS A3003 using GMAT Club Forum mobile app
Math Expert
Joined: 02 Aug 2009
Posts: 5545
Re: If x and y are positive integers greater than 3, what is the remainder [#permalink]

### Show Tags

06 Dec 2017, 09:52
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
_________________

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

BANGALORE/-

Intern
Joined: 03 Jun 2017
Posts: 15
Location: United Kingdom
Schools: Stanford '22
GMAT 1: 570 Q45 V23
GPA: 4
Re: If x and y are positive integers greater than 3, what is the remainder [#permalink]

### Show Tags

09 Dec 2017, 00:27
chetan2u wrote:
Bunuel wrote:

GMAT Club&#39;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?
Math Expert
Joined: 02 Aug 2009
Posts: 5545
Re: If x and y are positive integers greater than 3, what is the remainder [#permalink]

### Show Tags

09 Dec 2017, 03:33
1
KUDOS
Expert's post
stickman wrote:
chetan2u wrote:
Bunuel wrote:

GMAT Club&#39;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

BANGALORE/-

Intern
Joined: 03 Jun 2017
Posts: 15
Location: United Kingdom
Schools: Stanford '22
GMAT 1: 570 Q45 V23
GPA: 4
Re: If x and y are positive integers greater than 3, what is the remainder [#permalink]

### Show Tags

09 Dec 2017, 03:43
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?
Math Expert
Joined: 02 Aug 2009
Posts: 5545
If x and y are positive integers greater than 3, what is the remainder [#permalink]

### Show Tags

09 Dec 2017, 03:57
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

BANGALORE/-

Intern
Joined: 03 Jun 2017
Posts: 15
Location: United Kingdom
Schools: Stanford '22
GMAT 1: 570 Q45 V23
GPA: 4
If x and y are positive integers greater than 3, what is the remainder [#permalink]

### Show Tags

09 Dec 2017, 04:00
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.
Math Expert
Joined: 02 Sep 2009
Posts: 43381
Re: If x and y are positive integers greater than 3, what is the remainder [#permalink]

### Show Tags

09 Dec 2017, 04:07
1
KUDOS
Expert's post
1
This post was
BOOKMARKED
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.
_________________
Re: If x and y are positive integers greater than 3, what is the remainder   [#permalink] 09 Dec 2017, 04:07
Display posts from previous: Sort by