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# If a and b are positive integers such that a > b, what is the remainde

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Math Expert
Joined: 02 Sep 2009
Posts: 57298
If a and b are positive integers such that a > b, what is the remainde  [#permalink]

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06 Aug 2017, 00:55
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If a and b are positive integers such that a > b, what is the remainder when (a − b) is divided by 3?

(1) (2a − b) and (a + b), when divided by 9, leave remainder 6 in each case.
(2) (a + b) is divisible by 3.

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Re: If a and b are positive integers such that a > b, what is the remainde  [#permalink]

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06 Aug 2017, 04:34
1
Bunuel wrote:
If a and b are positive integers such that a > b, what is the remainder when (a − b) is divided by 3?

(1) (2a − b) and (a + b), when divided by 9, leave remainder 6 in each case.
(2) (a + b) is divisible by 3.

(1) 2a-b/a+b can be 15,24,33,42,51,60,.....
if a+b=15 then a=10, b=5
if a+b = 24 then a = 16, b=8
if a+b = 33 then a= 22, b=11
we can see the pattern
a=2b i.e. a-b = b in all cases b is divided by 3 leaving the remainder of 2
Sufficient

(2) a+b can be 3,6,9,12,15,18.........
if a+b=3 then a=2, b=1 and remainder is 1
if a+b=6 then a=4, b=2 and remainder is 2..Also a=5 and b=1 and remainder will be 1
Not Sufficient

A
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Re: If a and b are positive integers such that a > b, what is the remainde  [#permalink]

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15 Nov 2017, 14:35
3
Statement 1:
$$(2a - b) = 9x + 6$$ -----(1) where x is integer
$$(a + b) = 9y + 6$$ ------(2), where y is integer
Adding (1)+(2) => $$3a = 9(x+y) + 12 => a = 3(x+y) + 4$$

Multiplying ----(2) with 2 and subtracting from ----(1)
=>$$2a + 2b = 18y + 12$$ ---- (2) multiplied by 2
$$2a - b = 9x + 6$$ ----- (1)
now subtracting => $$3b = 18y - 9x + 6 => b = 6y - 3x + 2$$

so $$a - b = 3(x+y) + 4 - (6y - 3x + 2)$$
Re-arranging => $$3x + 3y + 4 - 6y + 3x - 2 => 6x - 3y + 2$$
from above,$$6x$$ is divisible by 3, $$3y$$ is divisible by 3, so remainder must be 2

Sufficient

Statement 2:
$$a = 7, b = 2 => (a + b)$$ is divisible by 3, but $$(a-b)$$ is not
$$a = 9, b = 3 =>$$ both $$(a+b)$$ and $$(a-b)$$ divisible by 3

Not Sufficient

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Re: If a and b are positive integers such that a > b, what is the remainde  [#permalink]

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08 Aug 2019, 05:28
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Re: If a and b are positive integers such that a > b, what is the remainde   [#permalink] 08 Aug 2019, 05:28
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# If a and b are positive integers such that a > b, what is the remainde

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