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

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Senior Manager
Joined: 22 Feb 2018
Posts: 425
If a and b are positive integers and a > b, what is the remainder when  [#permalink]

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14 Oct 2018, 07:49
4
00:00

Difficulty:

75% (hard)

Question Stats:

51% (02:07) correct 49% (01:36) wrong based on 55 sessions

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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 $$2$$.

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Re: If a and b are positive integers and a > b, what is the remainder when  [#permalink]

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14 Oct 2018, 08:11
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: 7335
Re: If a and b are positive integers and a > b, what is the remainder when  [#permalink]

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14 Oct 2018, 08:23
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
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html

GMAT Expert

Re: If a and b are positive integers and a > b, what is the remainder when   [#permalink] 14 Oct 2018, 08:23
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