GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 20 Aug 2018, 23:05

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# The positive integers are such that p < q r < s <

Author Message
TAGS:

### Hide Tags

Manager
Status: GMAT in 4 weeks
Joined: 28 Mar 2010
Posts: 166
GPA: 3.89
The positive integers are such that p < q r < s <  [#permalink]

### Show Tags

20 May 2011, 00:34
1
00:00

Difficulty:

65% (hard)

Question Stats:

40% (03:08) correct 60% (02:31) wrong based on 20 sessions

### HideShow timer Statistics

The positive integers are such that p < q ≤ r < s < 100, ps = qr and √s - √p ≤ 1. What is the value of p?

(X) The last digit of s is either 1, 2 or 3
(Y) 50 < p and r < 90

_________________

If you liked my post, please consider a Kudos for me. Thanks!

VP
Status: There is always something new !!
Affiliations: PMI,QAI Global,eXampleCG
Joined: 08 May 2009
Posts: 1124

### Show Tags

20 May 2011, 04:22
a s can be 1,2,3 means s can be 81 and p can be 64 giving s^(1/2) - p^(1/2) = 1
similarly s and p can have values such that s^(1/2) - p^(1/2) <= 1 decimal values possible.

b tells nothing about p and s relatively.

a+b where s = 81, p = 64 q and r = 72 each fits perfectly.
hence C.
_________________

Visit -- http://www.sustainable-sphere.com/
Promote Green Business,Sustainable Living and Green Earth !!

Manager
Status: GMAT in 4 weeks
Joined: 28 Mar 2010
Posts: 166
GPA: 3.89

### Show Tags

20 May 2011, 04:55
Solution:

Given that s-p > r -q as p < q <= r < s
=> (s-p)^2 > (r-q)^2
=> (s-p)^2 + 4sp > (r-q)^2 + 4rq [becuase sp = rq, we add 4sp in LHS and 4rq in RHS]
=> (s+p)^2 > (r+q)^2
=> s+p >= r+q+1 [as all numbers are integers] -> (1)

Suppose √s - √p = 1 (the other possibility is √s - √p < 1 that we will see later)
=> s + p - 2√sp = 1 => s+p = 1 + 2√qr [becuase sp = rq]
but (1) tells that √qr >= q + r => r = q [By AM-GM rule on positive numbers] and p+s = 2q + 1.

Now ps = q^2 => gcd(p, s) = 1 (the explanation is below)
Because if x divides gcd (p, s) and x is prime (or it will be product of two or more primes, but we assume the base case which covers other case as well), then x would divide q [because p = ax, q = bx => ps = abx^2 = q^2 and gcd(a, x) = 1 and gcd(b, x) = 1) and thus x dividing 2q+1 [= p+s = x(a+b)] is a contradiction => each of p and s is a perfect square [gcd(p, s) = 1].

If √s - √p < 1 then p + s < 1 + 2√ps <= 1 + q + r <= p + s which is a contradiction.

=> In all s and p are perfect squares. Now take X -> only possible s is 81 => p = 64
Now take Y, p > 50 => p can be 64 or 81 but if p = 81 then s = 100 (not possible as s < 100) => p can only be 64. The information on r is required to cross-check if our data in hand is correct and it indeed is as √64.81 = 72.
_________________

If you liked my post, please consider a Kudos for me. Thanks!

GMAT Tutor
Joined: 24 Jun 2008
Posts: 1345

### Show Tags

20 May 2011, 06:32
Where is this question from? It's one of the most unreasonable practice questions I've seen. Completing the proof alone takes well longer than two minutes, and it also requires repeated application of the Arithmetic Mean/Geometric Mean inequality, which is beyond the scope of the GMAT. Unless you're applying to do a Masters degree in Number Theory, you should ignore this question and work on more realistic practice material.

Not only that, but the OA posted is incorrect. If you do the work you find that the only sequences p, q, r, s which satisfy all of the given conditions are in the form x^2, (x)(x+1), (x)(x+1), (x+1)^2 where x is an integer. Since that's the case, each statement alone forces our sequence to be 64, 72, 72, 81, so the answer ought to be D, not C.
_________________

GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

Re: Number Properties &nbs [#permalink] 20 May 2011, 06:32
Display posts from previous: Sort by

# Events & Promotions

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.