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

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

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New post 14 Oct 2018, 07:49
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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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New post 14 Oct 2018, 08:11
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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.

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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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New post 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


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

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