If P and Q are positive integers, is P divisible by 12?

Question

(1) 17P + 24Q = 543
(2) P is eight more than seven times Q and Q is odd.

Are You Up For the Challenge: 700 Level Questions

Project DS Butler Data Sufficiency (DS3) 
 For DS butler Questions  Click Here

chetan2u · Accepted Answer

Is P=12a, where a is an integer?

(1) 17P + 24Q = 543
 17P=543-24Q 
 17P=(540+3)-24Q=540-24Q+3 
 17P=12*45-12*2Q+3 
 17P=12(45-2Q)+3 
So 17P is equal to a multiple of 12 plus 3. Hence P will never be a multiple of 12.
Also  17P=12(45-2Q)+3=even+odd=odd . P is odd.
Sufficient

(2) P is eight more than seven times Q and Q is odd.
P=7Q+8 and Q is ODD
 P=odd*7+8=odd*odd+even=odd+even=odd. 
Hence P is not divisible by 12.
Sufficient

D

TarunKumar1234 · Answer

If P and Q are positive integers, is P divisible by 12?

(1) 17P + 24Q = 543

St 1 is easiest to analyze

543 is odd and hence not divisible by 12
24Q is clearly divisible by 12

Now 17P + multiple of 12 = Not multiple of 12

Hence 17 P is NOT a multiple of 12
hence 12 is NOT a multiple of 12.

Andrewcoleman

GMATBusters there is an easier way to do statement 1

Posted from my mobile device

kntombat · Answer

chetan2u , I have one question.

17P+24Q = 543 is given.
I first tried to find the LCM of 17 and 12 and the only possible number is 204 or 17* multiples of 12. So 204+24Q=543; 543-204=339 and we know that 339 will not be divisible be 24.

even if we do 17*24= 408 and use in the equation 17P+24Q = 543, 408+24Q = 543; 543-408= 135, which is also not divisible by 12 or 24.

So can we use this method to say that p is not divisible by 12 and Statement 1 is Sufficient ?

kntombat · Answer

chetan2u ,  Bunuel ,  VeritasKarishma ,  nick1816  can someone help me with my doubt ?

chetan2u · Answer

Yes, you can do that but it is going to involve a bit of calculations.

17P+24Q=543......17P+even=odd....17P=odd-even=odd
so 17P is odd, and is not divisible by ANY even number.

Use number properties to your advantage.

KarishmaB · Answer

Simply use the concept of odd-even. Even if it doesn't strike you immediately, statement 2 should make you think of it. Also, you have been given such difficult numbers as 543 and 17. That itself is a clue that you are not required to do any calculations.

(1) 17P + 24Q = 543

I say that 543 is odd but 24Q is even because of 24. Then 17P MUST be odd. Hence P MUST be odd. Then P cannot be divisible by 12. Sufficient

(2) P is eight more than seven times Q and Q is odd.

Q is odd so 7Q will be odd too. 
P = 8 + 7Q = 8 + Odd = Odd
Then P cannot be divisible by 12.
Sufficient.

Answer (D)

kntombat · Answer

Thank you for the reply  VeritasKarishma  and  chetan2u  .

Andrewcoleman · Answer

GMATBusters  there is an easier way to do statement 1

Posted from my mobile device

shanayaanna · Answer

Considering that P is divisible by 12
P = 12x 
1. 17*12x + 24Q = 543
 12(17x + 2Q) = 543
17x + 2Q = 543/12 which is not an integer; Hence no, not possible (Both P and Q are +ve integers)

2. P = 8+7Q
Since Q is odd, 7Q will also be odd
Thus P will be odd and not divisible by 12

Both statements definitely show that P is not divisible by 12

GMATBusters · Answer

If P and Q are positive integers, is P divisible by 12?

(1) 17P + 24Q = 543

St 1 is easiest to analyze

543 is odd and hence not divisible by 12
24Q is clearly divisible by 12

Now 17P + multiple of 12 = Not multiple of 12

Hence 17 P is NOT a multiple of 12
hence 12 is NOT a multiple of 12.

GMATBusters · Answer

St 2) P is eight more than seven times Q and Q is odd.
P = 8 + Odd = odd
Since P is odd , It cannot be multiple of 12.

Hence St2 is sufficient too.

If P and Q are positive integers, is P divisible by 12?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

If P and Q are positive integers, is P divisible by 12?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards