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If x and y are integers, is xy + 1 divisible by 3?
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09 Dec 2014, 15:02
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Re: If x and y are integers, is xy + 1 divisible by 3?
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09 Dec 2014, 20:00
If x and y are integers, is xy + 1 divisible by 3? (1) When x is divided by 3, the remainder is 1 > x=3n+1, if x=1, then 1*y+1 when y=1 not div by 3 and when y=2 div by 3 => insufficient (2) When y is divided by 9, the remainder is 8 >y=9m+8, if y=1, then x*1+1 when x=1 not div by 9 and when x=8 div by 9 => insufficient (1) + (2) : x=3n+1 and y=9m+8 where m and n are integers=> xy+1=(3n+1)(9m+8)+1 =27mn+9m+24n+8+1 =27mn+9m+24n+9 => this is div by 3 always hence is sufficient => C Ans. C)
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Re: If x and y are integers, is xy + 1 divisible by 3?
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09 Dec 2014, 20:15
Given x and y are integers. We have to find out whether xy+1 is divisible by 3 ?
Statement 1: When x is divided by 3, the remainder is 1
Therefore, x can be represented as 3*a+1 ( where a is an integer) therefore, xy+1 => (3*a+1)*y+1
Since 3*a*y is divisible by 3, we have to find out whether (y+1) is divisible by 3.
Statement 2: When y is divided by 9, the remainder is 8. Clearly insufficient, since we don't know the value of x. But, we can say that y => 9*b+8 ( where b is an integer)
Combining 1) and 2), we can say y+1 = 9*b + 9 is divisible by 3. Hence the answer should be C



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Re: If x and y are integers, is xy + 1 divisible by 3?
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09 Dec 2014, 20:22
Question: remainder of (x*y +1)? (1) Not sufficient. Just tells that x could be equal to 4, 10, 13,.... but nothing about y. (2) Not sufficient. Just tell that y could be 17, 26, 35,...but no information about x.
(1)+(2) Sufficient. If x=10, y=26, then (10*26 + 1) = 261 >which is divisible by 3, so the remainder is 0 If x=4, y=35, then (4*35 + 1) = 141 > divisible by 3, so the remainder equals 0
Answer C



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Re: If x and y are integers, is xy + 1 divisible by 3?
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10 Dec 2014, 01:11
If x and y are integers, is xy + 1 divisible by 3?
(1) When x is divided by 3, the remainder is 1. (2) When y is divided by 9, the remainder is 8.
Statement 1 x = 3m+1; m is an integer no information on y, hence insufficient.
Statement 2 y= 9n +8; n is an integer no information on x, hence insufficient.
Combining statements (1) and (2) and substituting x and y as (3m+1) and (9n+8) xy +1 = (3m+1)(9n+8)+1 = 27mn+24m+9n+9 = 3*(9mn+8m+3n+3) > xy+1 is divisible by 3 Hence, both statements together are sufficient
Answer: C



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Re: If x and y are integers, is xy + 1 divisible by 3?
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24 Nov 2018, 21:47
Should be C, since: xy + 1 = (3ka+1)y = 3ka*y +y = 3kb + 1 + 9kc + 8 = 3kb + 9kc + 9 = multiple of 3. Hence C.
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Re: If x and y are integers, is xy + 1 divisible by 3?
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25 Nov 2018, 02:39
Bunuel wrote: Tough and Tricky questions: Remainders. If x and y are integers, is xy + 1 divisible by 3? (1) When x is divided by 3, the remainder is 1. (2) When y is divided by 9, the remainder is 8. Kudos for a correct solution. Statement One: If x is divided by 3, then x is even number, i.e. 4 (since no info is given on Y INSFFICIENT y can be 1 or 3 for example) Statement Two: When Y is divided by 9, remainder is 8, I.E. y = 9q+8 , so possible values for Y are 8, 17, 26, 35 etc (INSUFFICIENT) Combining both we know that x is 4, 7, 10 etc and y is 8, 17, 26, 35 etc. hence xy + 1 divisible by 3 C



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Re: If x and y are integers, is xy + 1 divisible by 3?
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09 Dec 2018, 10:15
My approach: Stem: If x and y are integers, is xy + 1 divisible by 3? –> notice that xy+1 is divisible by three if xy is even (but not iff it is even)
(1) When x is divided by 3, the remainder is 1. x=3q+1 –> not divisible by 3 –> y could be divisible by 3 or it could not be, producing two different answers –> Hence, NOT SUFFICIENT
(2) When y is divided by 9, the remainder is 8. –>Hence, the remainder when divided by 3 is also 8 –>However, x could be divisible by 3 or it could not be, producing two different answers –>Hence, NOT SUFFICIENT
(1) & (2) together —>From 1, we know that remainder is 1 —>From 2, we know that remainder is 8 —>Iff the remainder product of xy and the sum of the remainder product of xy and 1 add to 3, xy+1 must be divisible
HENCE, (remainder x) * (remainder y) + (1) = 8*1+1= 9, hence, xy+1 is always divisible by 3: C




Re: If x and y are integers, is xy + 1 divisible by 3? &nbs
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09 Dec 2018, 10:15






