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

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Math Expert
Joined: 02 Sep 2009
Posts: 52971
If x and y are positive integers greater than 3, what is the remainder  [#permalink]

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06 Dec 2017, 08:05
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85% (hard)

Question Stats:

45% (02:20) correct 55% (02:12) wrong based on 130 sessions

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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.

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Joined: 26 Oct 2017
Posts: 27
Re: If x and y are positive integers greater than 3, what is the remainder  [#permalink]

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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: 7334
Re: If x and y are positive integers greater than 3, what is the remainder  [#permalink]

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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
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.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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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: 7334
Re: If x and y are positive integers greater than 3, what is the remainder  [#permalink]

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09 Dec 2017, 03:33
1
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
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html

GMAT Expert

Intern
Joined: 03 Jun 2017
Posts: 49
Location: United Kingdom
GMAT 1: 610 Q47 V27
GMAT 2: 700 Q49 V34
GPA: 4
Re: If x and y are positive integers greater than 3, what is the remainder  [#permalink]

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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: 7334
If x and y are positive integers greater than 3, what is the remainder  [#permalink]

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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
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html

GMAT Expert

Intern
Joined: 03 Jun 2017
Posts: 49
Location: United Kingdom
GMAT 1: 610 Q47 V27
GMAT 2: 700 Q49 V34
GPA: 4
If x and y are positive integers greater than 3, what is the remainder  [#permalink]

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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: 52971
Re: If x and y are positive integers greater than 3, what is the remainder  [#permalink]

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09 Dec 2017, 04:07
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.
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Math Expert
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Posts: 52971
Re: If x and y are positive integers greater than 3, what is the remainder  [#permalink]

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24 Dec 2018, 01:50
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.

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

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25 Dec 2018, 21:33
1
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.

Statement 1) The positive difference between x and y is 2. Assume x=4, y=6. The remainder (24/9)=6. Assume x=5, y=7 Remainder(35/90)=8. Two different answers. Not Sufficient.

Statement 2) x and y are both primes numbers. Any prime integer can be expressed as 6*n+1 or 6*n-1. There are three possibilities for the product of x and y. They are (6*m+1)*(6*n-1), (6*m-1)*(6m-1) or (6m+1)*(6m+1){Both expressions are the same.} and (6*n-1)*(6*n-1). Possibility 1 (6*m+1)*(6*n-1) applies to all prime numbers. Taking m=3 and n=5, so xy= 19*29. The remainder (19*29/9)=2. For m=7 and n=5, the remainder (43*29/9)=5. Two answers, Not Sufficient. Possibility 2 explores when x=y= any prime number. So, the product will be a square of the prime number. e.g. 5^2/9, the remainder is 7 or 7^2/9, the remainder is 4. Two different answers. Not Sufficient. Possibility 3 explores when the prime numbers differ by 2. So, simplifying possibility 3, (6*n-1)*(6*n-1)=(36n^2-1). Remainder when (36n^2-1) is divided by 9 is -1 or (9-1)=8. It will always be -1 or 8 irrespective of n. So, in summary, only possibility 3 of 3 possibilities gives us an accurate answer. Statement 2 on whole is Insufficient.

(1)+(2)

This is what possibility 3 discussed in 2 above assumes that the difference between x and y is 2. Hence, Sufficient.

Option C is the correct answer.
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Re: If x and y are positive integers greater than 3, what is the remainder   [#permalink] 25 Dec 2018, 21:33
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