If p and q are positive integers, how many integers are

[VJesus12]

If p and q are positive integers, how many integers are larger than pq and smaller than p(q + 2)?

A. 3
B. p + 2
C. p - 2
D. 2p - 1
E. 2p + 1

The OA is D.

Can anyone one give me some help, please? I've got confused. How can I determine the correct answer?

[Sionainn@PrincetonReview]

Make this more concrete by plugging in some values for p and q. The problem stipulates that they need to be positive so let's make p = 2 and q = 3. Then pq = 6 and p(q + 2) = 10. The integers larger than pq and smaller than p(q + 2) are 7, 8 and 9. So there are 3 integers. The answer to the question with the number we chose then is 3. Now go to the answer choice and replace p with the value we chose (2) and see which expression gives you 3. Only A and D give a value of 3, so eliminate B, C and E.

To narrow down the choice between these two, try another set of numbers. Let's make p = 3 and q = 4. Then pq = 12 and p(q + 2) = 18. The integers larger than pq and smaller than p(q + 2) are 13, 14, 15, 16 and 17. So there are 5 integers. The answer to the question with the number we chose then is 5. Now go to the remaining answer choice and replace p with the value we chose (3) and see which expression gives you 5. Choice A no longer works, so the answer is D.

Let me know if you have any questions.

Take care,

Sionainn Marcoux
BA - Stanford University, MPP - Harvard University
Instructor, tutor for Princeton Review and Airbnb host
In other words a blend of Jamie Escalante from Stand and Deliver, Julie from The Love Boat, and Schneider the Super from One Day at a Time.

[Scott@TargetTestPrep]

If p and q are positive integers, how many integers are larger than pq and smaller than p(q + 2)?

A. 3
B. p + 2
C. p - 2
D. 2p - 1
E. 2p + 1

We are asked to determine the number of integers between pq and p(q + 2) = pq + 2p. The number of integers from pq to pq + 2p, inclusive, is:

(pq + 2p) - pq + 1 = 2p + 1

However, the integers between pq and pq + 2p must exclude pq and pq + 2p themselves; thus, the number of integers between pq and pq + 2p is:

2p + 1 - 2 = 2p - 1

Answer: D