Is the product pqr divisible by 12?

Target question: Is the product pqr divisible by 12?
At first glance, it LOOKS like the correct answer is C.
However, we are not told whether ALL of the numbers (p, q and r ) are integers.
Sure, statements 1 and 2 indirectly tell us that p and q are divisible by 3 and 4, but r may or may not be an integer.

Given this, we can go straight to.....

Statements 1 and 2 combined
Statement 1 tells us that p is a multiple of 3
Statement 2 tells us that q is a multiple of 4
There are many values of p, q and r that satisfy BOTH statements. Here are two:
Case a: p = 3, q = 4 and r = 1. In this case, pqr = (3)(4)(1) = 12, which is divisible by 12. So, the answer to the target question is YES, pqr IS divisible by 12
Case b: p = 3, q = 4 and r = 0.5. In this case, pqr = (3)(4)(0.5) = 6, which is not divisible by 12. So, the answer to the target question is NO, pqr is NOT divisible by 12
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

1) p=3k ..not sufficient
2)q=4r..not sufficient where k and r are integers ..
combining1 & 2 ..we have no idea about r ..if it is integer pqr will be divisible by 12 else not ..hence answer is E

p is a multiple of 3

Knowing only that p is a multiple of 3 is not sufficient to answer the question. If q and r both = 1, then pqr is not divisible by 12; however, if q and r both = 12, then pqr is divisible by 12. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

q is a multiple of 4

Knowing only that q is a multiple of 4 is not sufficient to answer the question. If p and r both = 1, then pqr is not divisible by 12; however, if p and r both = 12, then pqr is divisible by 12. Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using both statements we see that pq is a multiple of 12. However, we are given no information about the value of r. If r = 1/2, then pqr is not divisible by 12; however, if r = 1, then pqr is divisible by 12. Thus, we still do not have enough information to answer the question.

Answer: E

Important: Note that r is not given as an integer

(1) p is a multiple of 3

We need at least a multiple of 4
—> Insufficient

(2) q is a multiple of 4

We need at least a multiple of 3
—> Insufficient

Combining (1) & (2)

p*q is a multiple of 12. BUT we don’t know whether r is an integer or not

Case1: If r = 1/3 or 1/4 or 1/12 or 1/2
—> p*q*r will not be a multiple of 12

Case2: If r = 1 or 2 or 3 or any integer
—> p*q*r is a multiple of 12

IMO Option E

Pls Hit kudos if you like the solution

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