If x = n + 5, y = n + 3, and x, y and n are positive integers, what is

Question

If x = n + 5, y = n + 3, and x, y and n are positive integers, what is the remainder when xy is divided by 6?

(1) When n is divided by 6, the remainder is 3
(2) When n is divided by 3, there is no remainder

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yashikaaggarwal · Answer

If x = n + 5, y = n + 3, and x, y and n are positive integers,

(1) When n is divided by 6, the remainder is 3
n = 6d+3
where d is interger
n = 3,9,15,.......................
x = n + 5, y = n + 3
value of x and y will be (8,6),(14,12),(20,18)
the remainder when xy is divided by 6 is 0
8*6 , 14*12 , 20*18 all are divisible by 6
(Sufficient)

(2) When n is divided by 3, there is no remainder
n = 3,6,9,12.........................
x = n + 5, y = n + 3
value of x and y will be (8,6),(11,9),(14,12)
whose product is not divisible by 6 everytime.
(insufficient)

IMO A

Archit3110 · Answer

given If x = n + 5, y = n + 3, and x, y and n are positive integers
target what is the remainder when xy is divided by 6
#1
When n is divided by 6, the remainder is 3
n = 3,9,15,21
x=8,14,20 and y = 6,12,18
remainder is 0 when xy/6
sufficient
#2
When n is divided by 3, there is no remainder

n = 3,6,9,12,15
x= 11,17, 20; y = 9,20,18
yes and no 
insufficient
OPTION A

BrentGMATPrepNow · Answer

Target question :    What is the remainder when xy is divided by 6?  
This is a good candidate for  rephrasing the target question .

Given: x = n + 5, y = n + 3, and x, y and n are positive integers 
So, xy = (n + 5)(n + 3) = n² + 8n + 15

REPHRASED target question :    What is the remainder when n² + 8n + 15 is divided by 6?

Aside: the video below has tips on rephrasing the target question

Statement 1 : When n is divided by 6, the remainder is 3  
This tells us that  n is 3 greater than some multiple of 6 
We can write: n = 6j + 3 (for some integer j)

Now take the REPHRASED target question and replace n with 6j + 3 to get:   What is the remainder when (6j + 3)² + 8(6j + 3) + 15 is divided by 6?  
Now that's just focus on the expression: (6j + 3)² + 8(6j + 3) + 15
Expand: 36j² + 36j + 9 + 48j + 24 + 15
Simplify: 36j² + 84j + 48
Rewrite as:  6 (6j² + 14j + 8), which means the expression is divisible by 6
So, the answer to the REPHRASED target question is   when n² + 8n + 15 is divided by 6, the remainder is 0 
Since we can answer the  REPHRASED target question  with certainty, statement 1 is SUFFICIENT

Statement 2 : When n is divided by 3, there is no remainder 
This tells us that  n is some multiple of 3

Let's test some possible values of n:
 Case a : n = 3. In this case, n² + 8n + 15 = 3² + 8(3) + 15 = 9 + 24 + 15 = 48. So the answer to the REPHRASED target question is  when n² + 8n + 15 is divided by 6, the remainder is 0 
 Case b : n = 6. In this case, n² + 8n + 15 = 6² + 8(6) + 15 = 36 + 48 + 15 = 99. So the answer to the REPHRASED target question is  when n² + 8n + 15 is divided by 6, the remainder is 3 
Since we can’t answer the  REPHRASED target question  with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent
VIDEO ON REPHRASING THE TARGET QUESTION: 
 https://www.youtube.com/watch?v=MyG1K3ee69w

Ranasaymon · Answer

If x = n + 5, y = n + 3, and x, y and n are positive integers, 
What is the remainder of (xy)/6 ?
or, x*y/6=  {(n+5)*(n+3)}/6

(1) n/6 remainder 3, then n=3,9,15...
n=3,  {(3+5)*(3+3)}/6  , 
 (8*6)/6 Remainder 0
n=9,  {(9+5)*(9+3)}/6  , 
 (14*12)/6, remainder 0
st. 1 has only 1 value, so it is sufficient

(2) n/3 ,remainder 0, then n=3,6,9...
n=3,  {(3+5)*(3+3)}/6  , 
 (8*6)/6 Remainder 0
n=6,  {(6+5)*(6+3)}/6  , 
 (11*9)/6 Remainder 3
st. 2 has more than 1 value, so it is not sufficient

A is the answer

GMATWhizTeam · Answer

Solution 
 Step 1: Analyse Question Stem 
 • x, y, and n are positive integers.
• x = n + 5
• y = n + 3
 o xy = (n +5)(n+3) 
• We need to find the remainder when xy i.e. (n+5)(n+3) is divided by 6.

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1:  When n is divided by 6, the remainder is 3
 • According to this statement: n = 6k + 3, where k is a non-negative integer.
• Now, xy = (n+5)(n+3) = (6k + 3 + 5)(6k + 3 + 3) = (6k + 8)(6k + 6) = 6(6k + 8)(k +1)
 o Since, xy is a multiple of 6, therefore, when xy is divided by 6 the remainder will be 0.  
Hence, statement 1 is sufficient and we can eliminate answer Options B, C and E.

Statement 2:  When n is divided by 3, there is no remainder
 • According to this statement: n = 3m, where m is a positive integer.
• Now, xy = (n+5)(n+3) = (3m + 5)(3m + 3) = 3(3m+5)(m+1)
 o From the above the, we can only say that xy is a multiple of 3. We cannot be sure what will be the remainder when xy is divided by 6. For example:
  If m is odd, let’s say 1, then xy =3(3m+5)(m+1) =3*8*2, which is a multiple of 6
 • So, in this case, when xy is divided by 6, the remainder will be 0. 
 And if m is an even number, let’s say 2, then xy =3(3m+5)(m+1) =3*11*3, which is a not multiple of 6.
 • So, in this case, when xy is divided by 6 the remainder will be a non-zero number.   
• We are getting contradictory results. 
Hence statement 2 is not sufficient.

Thus, the correct answer is  Option A.

KaramveerBakshi · Answer

x = n+5 ; y = n+3
We need to find the remainder when xy/6

Statements:

(1) When n is divided by 6, the remainder is 3
This gives us n = 6k + 3
Therefore, x = 6k + 8 and y = 6k + 6
xy = (6k+8)(6k+6)
 = 2(3k+4)6(k+1)
 = 12(3k+4)(k+1)

xy is divisible by 12. ( And 6 too.)

So, the remainder when xy is divided by 6 = 0

Sufficient

(2) When n is divided by 3, there is no remainder
n = 3k
Therefore, x = 3k + 5 and y = 3k + 3
We have, xy = 3(3k+5)(k+1)

For k = 1, xy is not divisible by 6 and for k=2, xy is divisible by 6.

Insufficient

Hence, the answer is Option (A).

sambitspm · Answer

Kindly see the attachment. This question does take some time, especially for Option B.

Keep in mind that +ve integers mean that zero is not included. 0 is even integer but neither +ve nor -ve.

For option A, 
y = 6a + 6 = 6b
x = 6a + 8 = 6b+2

IMO A

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If x = n + 5, y = n + 3, and x, y and n are positive integers, what is

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