GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 02 Jun 2020, 08:00 ### 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

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # Let a, b, and c be positive integers with a> b > c such that a^2 - b^2

Author Message
TAGS:

### Hide Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 64172
Let a, b, and c be positive integers with a> b > c such that a^2 - b^2  [#permalink]

### Show Tags 00:00

Difficulty:   65% (hard)

Question Stats: 18% (02:12) correct 82% (03:29) wrong based on 11 sessions

### HideShow timer Statistics

Let a, b, and c be positive integers with a> b> c such that $$a^2 - b^2 - c^2 + ab = 2011$$ and $$a^2 + 3b^2 + 3c^2 - 3ab - 2ac - 2bc = -1997$$. What is a?

(A) 249
(B) 250
(C) 251
(D) 252
(E) 253

_________________
VP  V
Joined: 28 Jul 2016
Posts: 1020
Location: India
Concentration: Finance, Human Resources
Schools: ISB '18 (D)
GPA: 3.97
WE: Project Management (Investment Banking)
Re: Let a, b, and c be positive integers with a> b > c such that a^2 - b^2  [#permalink]

### Show Tags

Awaiting explanation by someone:
I was able to add two equations and come up with

$$a^2−b^2−c^2+ab=2011$$
$$a^2+3b^2+3c^2−3ab−2ac−2bc=−1997$$
------------------------------------------------
$$2a^2+2b^2+2c^2−2ab−2ac−2bc =14$$
or
$$a^2+b^2+c^2−ab−ac−bc =7$$

and
$$a^3+b^3+c^3−3abc=(a+b+c)[a^2+b^2+c^2−ab−bc−ac]$$

$$a^3+b^3+c^3−3abc=7(a+b+c)$$

But could not solve further
_________________

Keep it simple. Keep it blank
Intern  B
Joined: 18 Nov 2018
Posts: 27
Re: Let a, b, and c be positive integers with a> b > c such that a^2 - b^2  [#permalink]

### Show Tags

IMO E

Combine two equations and we have
a^2 + b^2 + c^2 - 2ab - 2ac - 2bc = 14
(a-b)^2 + (b-c)^2 + (a-c)^2 = 14

Since a>b>c and all three are integer, (a-c)^2 is 9 while (b-c)^2 and (a-b)^2 can be either 1 and 4.

Two scenarios: either a=c+3 & b=c+2 or a=c+3 & b=c+1

Replace the scenarios into the equation a^2 -b^2 -c^2 +ab = 2011 and we get only 1 integer solution for c which is 250. a=250+3=253.

Posted from my mobile device Re: Let a, b, and c be positive integers with a> b > c such that a^2 - b^2   [#permalink] 18 Apr 2019, 03:24

# Let a, b, and c be positive integers with a> b > c such that a^2 - b^2  