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If x and y are integers, is x + y 1 divisible by 3?

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Bunuel
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If x and y are integers, is x + y 1 divisible by 3? [#permalink] If x and y are integers, is x + y 1 divisible by 3?   11 May 2022, 09:00
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A
B
C
D
E

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If x and y are integers, is x + y − 1 divisible by 3?

(1) When x is divided by 3, the remainder is 2.

(2) When y is divided by 6, the remainder is 5.
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C
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Re: If x and y are integers, is x + y 1 divisible by 3? [#permalink] If x and y are integers, is x + y 1 divisible by 3?   11 May 2022, 09:27
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Bunuel wrote:
If x and y are integers, is x + y − 1 divisible by 3?

(1) When x is divided by 3, the remainder is 2.

(2) When y is divided by 6, the remainder is 5.

Both individually are INSUFFICIENT.

Together 1 and 2 they satisfy.
Note that the x and y leave a remainder 7(2+5) which when added(1 is subtracted) to -1 leaves 6 i.e. a multiple of 3.

Answer C.
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Re: If x and y are integers, is x + y 1 divisible by 3? [#permalink] If x and y are integers, is x + y 1 divisible by 3?   11 May 2022, 22:31
IMO C
When x is divided by 3, the remainder is 2.
=> x=3Q+2 --> Insufficient (A and D are out)
When y is divided by 6, the remainder is 5.
=> y=6P+5---> Insufficient (B is out)

Combining :
x+y-1= 3Q+2+6P+5-1
x+y-1= 3Q+6P+6
x=y-1= 3(Q+2P+2)
thus it is divisible by 3 we are getting a definite answer so C is enough. so E is out
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Re: If x and y are integers, is x + y 1 divisible by 3? [#permalink] If x and y are integers, is x + y 1 divisible by 3?   15 Nov 2022, 04:10
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Bunuel statements hold true for positive integers, but when we take a negative intergers, statement 1 & 2 combined are also insufficient.

e.g. x= -5
y= 17

x+y-1= -5+17-1 = 11, which is not divisible by 3. however positive values are.

let me know if there is something wrong with my reasoning.
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Re: If x and y are integers, is x + y 1 divisible by 3? [#permalink] If x and y are integers, is x + y 1 divisible by 3?   15 Nov 2022, 04:56
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Bunuel wrote:
If x and y are integers, is x + y − 1 divisible by 3?

(1) When x is divided by 3, the remainder is 2.

(2) When y is divided by 6, the remainder is 5.


Solution:
Pre Analysis:
  • We are asked if \(x+y-1=3k\) or \(x+y=3k+1\) where k is a positive integer

Statement 1: When x is divided by 3, the remainder is 2
  • No information on y
  • Thus, statement 1 alone is not sufficient and we can eliminate options A and D

Statement 2: When y is divided by 6, the remainder is 5
  • No information on x
  • Thus, statement 2 alone is also not sufficient

Combining:
  • Accoroding to statement 1, \(x=3p+2\)
  • According to statement 2, \(y=6q+5\)
  • \(x+y=3p+2+6q+5\)
    \(⇒x+y=3p+6q+7\)
    \(⇒x+y=3p+6q+6+1\)
    \(⇒x+y=3(p+2q+2)+1\) where \(p+2q+1\)\(\) is positive integer
  • This is exactly what we were asked

Hence the right answer is Option C
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Re: If x and y are integers, is x + y 1 divisible by 3? [#permalink]
15 Nov 2022, 04:56
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Bunuel
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