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

 It is currently 13 Oct 2019, 19:19

### 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

# If a and b are positive integers and a > b, what is the remainder when

Author Message
TAGS:

### Hide Tags

Senior Manager
Joined: 22 Feb 2018
Posts: 420
If a and b are positive integers and a > b, what is the remainder when  [#permalink]

### Show Tags

14 Oct 2018, 08:49
5
00:00

Difficulty:

75% (hard)

Question Stats:

51% (02:04) correct 49% (01:45) wrong based on 75 sessions

### HideShow timer Statistics

If $$a$$ and $$b$$ are positive integers and $$a > b$$, what is the remainder when $$a^2 - 2ab + b^2$$ is divided by $$9$$?

(1) The remainder when $$a-b$$ is divided by $$3$$ is $$2$$.
(2) The remainder when $$a-b$$ is divided by $$9$$ is $$2$$.

_________________
Good, good Let the kudos flow through you
Director
Joined: 19 Oct 2013
Posts: 520
Location: Kuwait
GPA: 3.2
WE: Engineering (Real Estate)
Re: If a and b are positive integers and a > b, what is the remainder when  [#permalink]

### Show Tags

14 Oct 2018, 09:11
1
Princ wrote:
If $$a$$ and $$b$$ are positive integers and $$a > b$$, what is the remainder when $$a^2 - 2ab + b^2$$ is divided by $$9$$?

(1) The remainder when $$a-b$$ is divided by $$3$$ is $$2$$.
(2) The remainder when $$a-b$$ is divided by $$9$$ is $$2$$.

From question stem it is asking for $$\frac{(a-b)^2}{9}$$

Statement 1) $$\frac{(a-b)}{3} = q + \frac{2}{3}$$ this can be converted to $$a-b = 3q + 2$$

if q = 0 then a-b = 2 and $$(a-b)^2$$ = 4 as a result it is 4/9

if q = 1 then a-b = 5 and $$(a-b)^2$$ = 25 as a result it is $$\frac{25}{9} = 2 \frac{7}{9}$$

Insufficient.

Statement 2) $$\frac{(a-b)}{9} = q + \frac{2}{9}$$ this can be converted to $$a-b = 9q + 2$$

if q = 0 then a-b = 2 and $$(a-b)^2$$ = 4 as a result it is 4/9

if q = 1 then a-b = 11 and $$(a-b)^2$$ = 121 as a result it is $$\frac{121}{9} = 13 \frac{4}{9}$$

Sufficient.

Math Expert
Joined: 02 Aug 2009
Posts: 7943
Re: If a and b are positive integers and a > b, what is the remainder when  [#permalink]

### Show Tags

14 Oct 2018, 09:23
If $$a$$ and $$b$$ are positive integers and $$a > b$$, what is the remainder when $$a^2 - 2ab + b^2$$ is divided by $$9$$?
Now $$a^2 - 2ab + b^2=(a-b)^2$$
(1) The remainder when $$a-b$$ is divided by $$3$$ is $$2$$.
cases when reaminder is 2 is when
a-b=2, remainder of 4 when 2^2 is divided by 9
a-b=5, remainder of 7 when 5^2 is divided by 9
a-b=8, remainder of 1 when 8^2 is divided by 9
insuff

(2) The remainder when $$a-b$$ is divided by $$9$$ is $$2$$.
so (a-b)62 will leave a remainder of 2^2=4
suff

B
_________________
Manager
Joined: 29 Dec 2018
Posts: 81
Location: India
WE: Marketing (Real Estate)
Re: If a and b are positive integers and a > b, what is the remainder when  [#permalink]

### Show Tags

14 Jun 2019, 19:50
Dear Salsanousi , chetan2u

I read both your explanations but did not really understand why statement 2 is sufficient, could you please explain in more simplistic manner?

(May be I cannot visualize the statement 2 - remainder is 2 always)
_________________
Keep your eyes on the prize: 750
Director
Joined: 20 Jul 2017
Posts: 844
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)
Re: If a and b are positive integers and a > b, what is the remainder when  [#permalink]

### Show Tags

14 Jun 2019, 20:11
1
Princ wrote:
If $$a$$ and $$b$$ are positive integers and $$a > b$$, what is the remainder when $$a^2 - 2ab + b^2$$ is divided by $$9$$?

(1) The remainder when $$a-b$$ is divided by $$3$$ is $$2$$.
(2) The remainder when $$a-b$$ is divided by $$9$$ is $$2$$.

a^2 - 2ab + b^2 = (a - b)^2

(1) a - b = 3k + 2, for any integer k
—> (a - b)^2 = (3k + 2)^2 = 9k^2 + 12k + 4
When divided by 9
—> (9k^2 + 12k + 4)/9 = k^2 + (12k + 4)/9
Many remainders are possible.
—> No unique value - Insufficient

(2) a - b = 9m + 2, for any integer m
—> (a - b)^2 = (9m + 2)^2 = 81m^2 + 36m + 4
When divided by 9
—> (81m^2 + 36m + 4)/9
—> 9m^2 + 4m + 4/9
—> A unique remainder = 4
Sufficient

IMO Option B

Pls Hit kudos if you like the solution

Posted from my mobile device
_________________
Pls Hit Kudos if you like the solution
Director
Joined: 24 Nov 2016
Posts: 559
Location: United States
If a and b are positive integers and a > b, what is the remainder when  [#permalink]

### Show Tags

08 Oct 2019, 04:35
Princ wrote:
If $$a$$ and $$b$$ are positive integers and $$a > b$$, what is the remainder when $$a^2 - 2ab + b^2$$ is divided by $$9$$?

(1) The remainder when $$a-b$$ is divided by $$3$$ is $$2$$.
(2) The remainder when $$a-b$$ is divided by $$9$$ is $$2$$.

(1) The remainder when $$a-b$$ is divided by $$3$$ is $$2$$: insufic.
$$remainder:\frac{(a-b)^2}{9}…\frac{(3m+2)^2}{9}…\frac{9m+4+12m}{9}…\frac{4+12m}{9}$$
$$m=0:\frac{4}{9}…remainder=4$$
$$m=1:\frac{4+12}{9}=\frac{16}{9}…remainder=7$$

(2) The remainder when $$a-b$$ is divided by $$9$$ is $$2$$: sufic.
$$remainder:\frac{(a-b)^2}{9}…\frac{(9n+2)^2}{9}…\frac{9n+4+36n}{9}…\frac{4}{9}$$
$$remainder=4$$

If a and b are positive integers and a > b, what is the remainder when   [#permalink] 08 Oct 2019, 04:35
Display posts from previous: Sort by