BTGmoderatorLU wrote:Source: e-GMAT

The number \(A\) can be expressed as \(p*q\) where \(p\) and \(q\) are positive integers. Is \(A\) divisible by 16?

1. \(p=8*k\), where \(k\) is an odd number.
2. \(q^2-8q+15=0.\)

The OA is C

We have A = pq.

We need to determine whether A is divisible by 16; for it to happen pq must be divisible by 16.

Let's take each statement one by one.

1. \(p=8*k\), where \(k\) is an odd number.

pq = 8kq

If q is even, the answer is yes; however, if q is odd, the answer is no. Insufficient.

2. \(q^2-8q+15=0.\)

q^2 - 3q - 5q + 15 = 0
q(q - 3) - 5(q - 3) = 0
q = 3 or 5, odd numbers

If p itself is divisible by 16, the answer is yes; however, if it is not, the answer is no. Insufficient.

(1) and (2) together

So, from (1) and (2), we have

pq = 8*k*3 = 24k; where k is odd => we see that pq is not divisible by 16. The answer is no.

OR

pq = 8*k*5 = 40k; where k is odd => we see that pq is not divisible by 16. The answer is still no.

Sufficient

The correct answer: C

Hope this helps!

-Jay
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p and q almost always represent prime numbers in math, so I don't like their choice of letters here. Statement 1 tells us p is divisible by 8, but not by 16. So for pq to be divisible by 16, we need q to be even, something we don't know, so Statement 1 is not sufficient alone. Statement 2 gives a quadratic which clearly has only odd roots, since the roots multiply to 15, so q can only be odd. With no information about p, that's not sufficient, but with both statements we know pq cannot be divisible by 16, so the answer is C.