Hi, there. I'm happy to help with this --- it looks like a cool question.
Prompt: if r and s are integers, is r+s divisible by 3?
There are two ways to add up two integers and get a sum that's divisible by 3.
(a) both r and s individually are multiples of 3, so their sum will be a multiple of 3. (Example: 9 + 6 = 15)
(b) let's say that neither r nor s is divisible by 3; let's say that when we divide r by 3, the remainder is m, and when we divide s by 3, the remainder is n; if m + n = 3, then r + s will be a multiple of 3.
Example:
When 5 is divided by 3, remainder = 2
When 7 is divided by 3, remainder = 1
2+1 = 3 ----> 5 + 7 = 12
The basic idea there is that the sum of the remainders will be the remainder of the sum. If sum of the remainders is divisible by 3, then when the sum of the numbers is divided by three, it will go into the sum of the remainders evenly, and therefore go into the entire sum evenly.
BTW,
in everything I've said between the prompt and this line, you could replace 3 with any other positive integer greater than 1, and it would still be true. If we're in one of these two possibilities, then we will know that r+s is divisible by 3.
Statement #1: s is divisible by 3
Promising, but we know nothing about r, so this statement, by itself, is
insufficient.
Statement #2: r is not divisible by 3
Again, this could be a step in the right direction, but now we know nothing about s, so this statement, by itself, is
insufficient.
Combined Statements #1 & #2: s is divisible by 3 and r is not divisible by 3
Well, neither of the scenarios above allow for one number divisible by 3 and the other not divisible by 3. In fact, when we divide (r+s) by 3, we know the s part will not have a remainder but the r part will, which means that 3 does not go evenly into the sum r+s. Therefore, we can give a definitive "no" answer to the prompt question, and because we have the ability to give a definitive answer, that means we must have
sufficient information.
Statements are insufficient individually but are sufficient combined ==> Answer =
C.
Does all that make sense?
Here's a particularly difficult divisibility DS question, for more practice.
http://gmat.magoosh.com/questions/871Let me know if you have any questions about anything I have said here.
Mike
_________________
Mike McGarry
Magoosh Test PrepEducation is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)