If r and s are integers, is r+s divisible by 3?

If r and s are integers, is r+s divisible by 3?

(1) s is divisible by 3 --> s={a multiple of 3}. Not sufficient as no info about r.
(2) r is not divisible by 3 --> r={NOT a multiple of 3}. Not sufficient as no info about s.

(1)+(2) r+s={NOT multiple of 3}+{a multiple of 3}={NOT multiple of 3}. Sufficient.

Answer: C.

Below might help to understand this concept better.

If integers \(a\) and \(b\) are both multiples of some integer \(k>1\) (divisible by \(k\)), then their sum and difference will also be a multiple of \(k\) (divisible by \(k\)):
Example: \(a=6\) and \(b=9\), both divisible by 3 ---> \(a+b=15\) and \(a-b=-3\), again both divisible by 3.

If out of integers \(a\) and \(b\) one is a multiple of some integer \(k>1\) and another is not, then their sum and difference will NOT be a multiple of \(k\) (divisible by \(k\)):
Example: \(a=6\), divisible by 3 and \(b=5\), not divisible by 3 ---> \(a+b=11\) and \(a-b=1\), neither is divisible by 3.

If integers \(a\) and \(b\) both are NOT multiples of some integer \(k>1\) (divisible by \(k\)), then their sum and difference may or may not be a multiple of \(k\) (divisible by \(k\)):
Example: \(a=5\) and \(b=4\), neither is divisible by 3 ---> \(a+b=9\), is divisible by 3 and \(a-b=1\), is not divisible by 3;
OR: \(a=6\) and \(b=3\), neither is divisible by 5 ---> \(a+b=9\) and \(a-b=3\), neither is divisible by 5;
OR: \(a=2\) and \(b=2\), neither is divisible by 4 ---> \(a+b=4\) and \(a-b=0\), both are divisible by 4.

Hope it's clear.
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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.

https://gmat.magoosh.com/questions/871

Let me know if you have any questions about anything I have said here.

Mike
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I thought the answer will be E because r could be negative 3, and s can be 3. As a result we get -3+3/3= which is not divisible by 3?
I'm confused. Can someone help me to explain it?

Welcome to GMAT Club. Below is an answer to your question.

The red parts in your post are not correct.

1. r cannot be -3, since we are told that "r is NOT divisible by 3", while -3 is clearly divisible by 3. Note that integer \(a\) is divisible by integer \(b\) (integer \(a\) is a multiple of integer \(b\)) means that \(\frac{a}{b}=integer\), so as -3/3=-1=integer then -3 is divisible by 3.

2. Also if r=-3 (if r could be -3) and s=3 then r+s=-3+3=0 and zero is divisible by every integer except zero itself.

Hope it's clear.
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Got it! Thank you very much!

Seriously couldn't have said it better

The concept in this question is basically testing three fundamental scenarios

Scenario 1

X + Y / some multiple K- in order for the sum of X and Y to be a multiple of K both X and Y must be a multiple of K

Scenario 2

X + Y/ some multiple K- if either X or Y is not a multiple of K and the other is ...then the sum of X and Y cannot be a multiple of k

Scenario 2

X +Y/ some multiple K - Actually, the sum could be a multiple of K but not doesn't necessarily have to

C

Both the statements individually are not sufficiently because they don't say anything about both the variable.
Combining both the statements
S =3x
Where x is an integer
R = 3y+1 or 3y+2
Where y is an integer
Now S + R = 3(x+y) + 1 or 3(x+y) +2
In both the cases s+r is not divisible by 3. Hence together given statements are sufficient
Answer is option C

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