Brent@GMATPrepNow wrote: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

Answer: A
Difficulty level: 550 - 600
Source: www.gmatprepnow.com

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

Cheers,
Brent