If p and q are integers, is p + q divisible by 3?

Question

(1) p - q is divisible by 3.

(2) pq is divisible by 3.

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yashikaaggarwal · Answer

(1) p - q is divisible by 3.
P and Q have 3n no. difference between them
Let P,Q = (4,1) , (5,2) , (6,3)...............................
all have difference of 3 in between which is divisible by 3
But not all pairs sum is divisible by 3 is (4+1=5 is not divisible by 3)
(Insufficient)

(2) pq is divisible by 3.
either of the P and Q is a multiple of 3
let say Q = 3n 
when P= 1 , 1 + 3n will never be divisible by 3, but 3+3n is 
(Insufficient)

Combining both.
only possible vale of P and Q whose sum is divisible by 3 is when both of them are multiple of 3.
(Sufficient)

IMO C

GMATinsight · Answer

Question : is p + q divisible by 3?

Statement 1 : p - q is divisible by 3.

Case 1: p = 4 and q = 1, p-q = 3, p+q = 5 NO
Case 2: p = 6 and q = 3, p-q = 3, p+q = 9 YES

NOT SUFFICIENT

STatement 2 : pq is divisible by 3

i.e. atleast one of p and q is divisible by 3

Case 1: p = 3 and q = 1, p+q = 4 NO
Case 2: p = 6 and q = 3, p+q = 9 YES

NOT SUFFICIENT

COmbining the statements

p-q = 3a (multiple of 3)

One of p and q is also a multiple of 3 i.e. let p = 3b

since, p-q = 3b-q = 3a

i.e. q = 3(b-a)

i.e. p and q both are divisible by 3

then p+q = 3b + 3(b-a) = 3(2b-a) = Multiple of 3 YES

SUFFICIENT

Answer: Option C

hiranmay · Answer

If p and q are integers, is p + q divisible by 3?

(1) p - q is divisible by 3. -->  insuff: if p =4 & q=1, then the answer is NO, but if p =6 & q=3, then the answer is YES

(2) pq is divisible by 3.-->  insuff: if p =3 & q=1, then the answer is NO, but if p =3 & q=6, then the answer is YES

Combining (1) & (2)=>
let's say p - q = 3n => p= 3n+q
& pq = 3m i.e. at least of of them is multiple of 3, let's say q = 3s
p+q = 3n+q+q=3n+2q=3n+2*3s=3(n+2s)--> multiple of 3, we can prove the same assuming p=3t also
So the  answer: C

KaramveerBakshi · Answer

p, q are integers.
Q: Is (p+q)/3 = I or not?

Statements:

(1) p - q is divisible by 3.
p - q = 3k
or p = 3k + q
Now, p + q = 3k + 2q --- (A)

(A) is divisible by 3 if q=2 and not if q=1.

Insufficient.

(2) pq is divisible by 3.
pq=3k
p+q = p + p/3k. ---- (B)

(B) is divisible be 3 if p = 3n and not if p = 3n+1 or 3n+2

Insufficient.

Combining both statements,
Let's assume q=3m ( either q or p is divisible by 3 )
Now, p - q = 3k
We have p = 3k + 3m

Therefore, p+q = (3k+3m) + 3m
p+q is therefore divisible by 3.

Sufficient

Hence, the answer is Option(C)

sambitspm · Answer

Kindly see the attachment.
 Concept: Numbers given are integers and the ability to express numbers as multiples of others.

bumpbot · Answer

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If p and q are integers, is p + q divisible by 3?

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