Integers

[deepoe]

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

I chose P = 2 and Q = 4, So I came to answer A.

But the answer says:

Let p = 1 and q = 2. Then pq = 2 and p(q + 2) = 4. This scenario has one integer, 3, greater than pq
and less than p(q + 2). Now, we plug p = 1 and q = 2 into the answer-choices until we find one
that has the value 1. Look at choice (D): 2p – 1 = (2)(1) – 1 = 1. Thus, the answer is (D).


Do I need to choose 1 and 2 as positive integer next time ( but 1 is odd :S )

[DanaJ]

Notice that p(q + 2) = pq + 2p. You are looking for integers greater than pq and the immediately greater integer will be pq + 1. You're also looking for integers smaller than pq + 2p, and the immediately smaller integer will be pq + 2p - 1.
Now, remember that rule that states that the number of integers between m and n inclusively is (n - m) + 1. Use that to find the total number of integers:
(pq + 2p - 1 - pq - 1) + 1 = 2p - 1.

[cramya]

Use Number picking strategy.

2 rules to follow when picking numbers:

1) Dont pick numbers that appear in the answer
2) Dont pick the same number for 2 different variables

p=5 q=7

Only D will fit the bill

p=6 q-8 only D will fit the bill

[sureshbala]

Number of integers from A to B = B-A+1 (where A<B)
Number of integers between A and B = B-A-1 (where A<B)

Coming to our question, we need to find the number of integers between pq and pq+2p = (pq+2p)-pq-1 = 2p-1

[KICKGMATASS123]

why is it pq+2p-1 minus pq+1

why not the other way around?

[vittalgmat]

why is it pq+2p-1 minus pq+1

why not the other way around?

As Dana, Suresh and Cramya have explained, # of integers between A and B (not including both A and B) is B -A -1. Using this concept.
# of integers between pq and p (q+2) =
p(q+2) - pq -1
= pq +2p -pq -1
= 2p -1.

HT Helps
-V