Data Sufficiency

Is (p^2qr^3s + p^2qrs^3) / p^2qrs divisible by 8

a) Each of p,q,r,s, is divisible by 2 but not by 4.

b) Each of p,q,r is divisible by 2 but not by 4 and s is an odd integer.

1. statement (A) alone is sufficient but statement (B) alone is not
2. statement (B) alone is sufficient but statement (A) alone is not
3. both (A) and (B) together are sufficient but none of them alone is sufficient
4. both statements are sufficient independently
5. both (A) and (B) together are not sufficient

If the above equation is resolved, it comes to r^2 + S ^2 divisible by 8. How do we proceed further ?

govind_raj_76 wrote:Is (p^2qr^3s + p^2qrs^3) / p^2qrs divisible by 8

a) Each of p,q,r,s, is divisible by 2 but not by 4.

b) Each of p,q,r is divisible by 2 but not by 4 and s is an odd integer.

1. statement (A) alone is sufficient but statement (B) alone is not
2. statement (B) alone is sufficient but statement (A) alone is not
3. both (A) and (B) together are sufficient but none of them alone is sufficient
4. both statements are sufficient independently
5. both (A) and (B) together are not sufficient

If the above equation is resolved, it comes to r^2 + S ^2 divisible by 8. How do we proceed further ?

Before diving in, I'm going to assume (something we'd never do on the actual GMAT, but something we have to sometimes do for "made up" questions) that none of p, q, r or s can equal 0; if one of them can equal 0, then the question is flawed. (Is an undefined expression divisible by 8? I have no clue! You would never encounter that on the actual GMAT.)

Well, the question is flawed anyway, since the two statements contradict. Statement (1) says that s is divisible by 2 (i.e. s is even) and statement (2) says that s is odd. Since there are no numbers that are simultaneously odd and even, according to this question there's no possible value for s (at least in our universe).

On the actual GMAT, the two statements NEVER contradict each other.

All that said, I had so much fun with statement (1) that I'm going to provide a solution anyway!

Step 1 of the Kaplan Method for DS: Analyze the Stem

Let's start by simplifying the question.

Breaking it up into two fractions:

Is (p^2qr^3s/p^2qrs) + (p^2qrs^3/p^2qrs) divisible by 8?

Cancelling out as much as we can:

Is r^2 + s^2 divisible by 8?

Step 2 of the Kaplan Method for DS: Evaluate the Statements

(1) Each of p,q,r,s, is divisible by 2 but not by 4.

If r and s are both divisible by 2 but not 4, then r^2 and s^2 are each divisible by 4 but not 8.

So, possible values for r and s are: 2, 6, 10, 14, ...

and possible values for r^2 and s^2 are 4, 36, 100, 196, ...

You can experiment all you want, when you add up two of those numbers you ALWAYS get a multiple of 8: sufficient.

Here's how we could prove algebraically (if we really wanted to!):

Since neither r nor s is a multiple of 4, each one has exactly 1 factor of 2. Similarly, r^2 and s^2 each has exactly 2 factors of 2 (or 1 factor of 4).

In other words, we can reduce them to:

r = 2*odd
s = 2*odd

and:

r^2 = 4*odd
s^2 = 4*odd

Looking at our sum:

r^2 + s^2 = 4*odd + 4*odd = 4(odd + odd) = 4*even which will always be divisible by 8... hooray!

(2) Each of p,q,r is divisible by 2 but not by 4 and s is an odd integer.

If r is even and s is odd, then:

r^2 + s^2 = even + odd = odd which is definitely NOT a multiple of 8. Sufficient.

So, in theory the answer is (D); however, since the two statements contradict each other (for a yes/no question you'll never have one statement result in "definitely yes" and the other result in "definitely no"), this question is inherently flawed (although an interesting exercise).

What's the source?