When positive integer N is divided by 18, the remainder is x.

Question

When positive integer N is divided by 18, the remainder is x. When N is divided by 6, the remainder is y. Which of the following are possible values of x and y?

i) x = 9 and y = 3
ii) x = 16 and y = 2
iii) x = 13 and y = 7

A) i only
B) i and ii only
C) i and iii only
D) ii and iii only
E) i,ii and iii

GyMrAT · Accepted Answer

Given, N>0
N = 18k + x
N = 6q + y

Therefore, 18k + x = 6q + y

or x = 6(q-3k) + y

x = Multiple of 6 + y

Check the given statements
i) x = 9 and y = 3, possible for 9 = 6(1) + 3
ii) x = 16 and y = 2, not possible 16 = 6(2) + 4, given y is not equal to 4
iii) x = 13 and y = 7, not possible 13 = 6(2) + 1, given y is not equal to 1.

Hence only i) is true

Answer A.

Thanks,
GyM

SSM700plus · Answer

If N is the number divided 18 and leaves remainder x. 
It can be represented as N= 18p + x. Note x < 18. It can't be greater than or equal to 18. x is remainder and remainder is always less than divisor

If N is the number divided 6 and leaves remainder y.
it can be represented as N = 6q + y. Note y < 6. It can't be greater than or equal to 6. y is remainder and remainder is always less than divisor.

Also Note that if N is divisible by 18 then it will be divisible by 6 also. Only concern is what is the remainder if x is divisible by 6.

Well remainder is same as Y.

x= 6k +y.

x -y =6k.

difference between x & y is always multiple of 6. And x is less than 18 and y is less than 6. As mentioned earlier.

i) x = 9 and y = 3............Yes satisfies.
ii) x = 16 and y = 2...............Not satisfies. Difference not satisfied
iii) x = 13 and y = 7...............Not satisfies. y should be less than 6.

Hence Ans - A

rakeshmishra5690 · Answer

IMO A

we will have 3 cases

if 1<=x<=5 then x = y
if 6<=x<=11 then x = 6+y
id 12<=x<=17 then x = 12 +y

only statement 1 satisfies this

Hence Ans A

BrentGMATPrepNow · Answer

Let's examine each statement separately...

i) x = 9 and y = 3  
Let's come up with a value of N that satisfies this condition. 
How about N = 9?
9 divided by 18 = 0 with remainder 9 (i.e., x = 9)
...and 9 divided by 6 = 1 with remainder 3 (i.e., y = 3)
Perfect,  statement i is TRUE 
Check the answer choices.....ELIMINATE D

ii) x = 16 and y = 2  
Can you come up with a value of N that satisfies this condition?
How about N = 16?
16 divided by 18 = 0 with remainder 16 (i.e., x = 16, which WORKS)
However, 16 divided by 6 = 2 with remainder 4 (i.e., y = 4. NO GOOD)

How about N = 34?
34 divided by 18 = 1 with remainder 16 (i.e., x = 16, which WORKS)
However, 34 divided by 6 = 5 with remainder 4 (i.e., y = 4. NO GOOD)

We can keep testing N-values until we convince ourselves that there are no values of N that makes those values (x = 16 and y = 2) possible. 
So,  statement ii is FALSE 
Check the answer choices.....ELIMINATE B and E

HOWEVER, if you need more convincing that statement ii is FALSE, we can make the following observations:
 When positive integer N is divided by 18, the remainder is x : so, we can say that N = 18k + x for some integer k
 When positive integer N is divided by 6, the remainder is y : so, we can say that N = 6j + y for some integer j
We can combine the two equations to get: 18k + x = 6j + y
Isolate x to get: x = y + 6j - 18k
Factor right side to get: x = y +  6(j - 3k) 
Rewrite as: x = y +  some multiple of 6 
This means we'll never have the case where x = 16 and y = 2, because 16 CANNOT be written as 2 +  some multiple of 6 
So,  statement ii is FALSE

iii) x = 13 and y = 7  
We must be careful with this one. 
While it is true that 13 CAN be written as 7 +  some multiple of 6 , we must also consider the following property of remainders: 
  When positive integer N is divided by positive integer D, the remainder R is such that 0 ≤ R < D  
For example, if we divide some positive integer by 7, the remainder will be 6, 5, 4, 3, 2, 1, or 0

Based on the above   property  , when we divide N by 6, the remainder can be 5, 4, 3, 2, 1, or 0
So, the remainder CANNOT be 7
In other words, y CANNOT equal 7 
So,  statement iii is FALSE 
ELIMINATE C

Answer: A

TheNightKing · Answer

y=7 is straight up not Possible. because y can be <=6. 
So Option C,D, and E are out.

Let's evaluate (ii)
x=16

Let's say N=16. 16/18 Remainder=16. But 16/6 Remainder=4. So y is not equal to given value of 2. So Not Possible.

Hence A.

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When positive integer N is divided by 18, the remainder is x.

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