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

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

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

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.      
(D) EACH statement ALONE is sufficient.        
(E) Statements (1) and (2) TOGETHER are not sufficient.

“If P and Q are positive integers, is P divisible by 12?” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation:
Approach Solution 1:

The given case question is based on the aspect of P and Q being positive integers and finding whether P is divisible 12 or not. Accordingly, in order to solve the question proving if both the given statement conditions are true and sufficient, both need to be evaluated. 

1. The first statement evaluates that- 17P + 24Q = 543

The equation needs to be solved as follows: 

17P=543−24Q
17P=543−24Q
17P=(540+3)−24Q=540−24Q+3
17P=(540+3)−24Q=540−24Q+3
17P=12∗45−12∗2Q+3
17P=12∗45−12∗2Q+3
17P=12(45−2Q)+3
17P=12(45−2Q)+3

Based on this evaluation, it can be stated that 17P is a multiple of 12 plus 3 and thus, P will never be divisible by 12. Furthermore, another important evaluation needs to be considered- 
Also 
17P=12(45−2Q)+3=even+odd=odd
17P=12(45−2Q)+3=even+odd=odd. P is odd.

Accordingly, considering the fact from the equation above that P is odd. Hence, being an even number, P will not be divisible by 12 and hence, is sufficient in proving the same. 

2. The second statement states that P is eight more than seven times Q and Q is odd. In order to prove this, the following can be evaluated- 

Considering that Q is odd, it can be stated that- 

P=7Q+8 

P=odd∗7+8 
=odd∗odd+even 
= odd+even
= odd.

P=odd∗7+8
=odd∗odd+even
=odd+even
=odd.

While Q was already stated to be an odd number, the equation was able to prove that P is also an odd number. Thus, P cannot be divisible by 12 and hence, this answer is also sufficient for the question. 

Correct Answer: D

Approach Solution 2:

Based on the given question that needs to find if P is divisible by 12 or not, both the given statement conditions need to be proved sufficient enough. This is mainly to prove if P is divisible by 12 or not divisible by 12. 

According to the statement 1, 17P + 24Q = 543, to solve the question, the given equation needs to be solved. Then the resultant unique value comes out for P equalling to 15 and Q as 12. Hence, P is not divisible by 12 and thus, sufficient. 

Based on the second given statement, P is eight more than seven times Q and Q is odd. 
According to this statement, it can be evaluated that- 
P-7Q =8, Q=1, positive and odd integer. 
This implies that P = 8+7 = 15

Thus, with the value of P being 15, it can be evaluated that P is not divisible by 12. Hence, this statement is sufficient. 

Correct Answer: D

Approach Solution 3:

let us check both statements one at a time.

Statement 1: 

17P + 24Q = 543
We have unique value of P= 15 and Q= 12 and P is not divisible by 12. 

Hence, the statement is Sufficient.

Statement 2: 

P is eight more than seven times Q and Q is odd.
P-7Q =8, Q=1, positive and odd integer and P = 8+7 = 15, is not divisible by 12. 

Hence, this statement is also Sufficient.

Correct Answer: D

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