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Senior Manager  Status: Do and Die!!
Joined: 15 Sep 2010
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If x, y and k are integers, is xy divisible by 3? (1) y =  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 34% (01:56) correct 66% (02:05) wrong based on 171 sessions

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If x, y and k are integers, is xy divisible by 3?

(1) y = 2^(16) - 1

(2) The sum of the digits of x equals 6^k

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Originally posted by shrive555 on 24 Oct 2010, 12:34.
Last edited by shrive555 on 24 Oct 2010, 12:54, edited 1 time in total.
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Re: integers divisibility  [#permalink]

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shrive555 wrote:
If x, y and k are integers, is xy divisible by 3?

(1) y = 216 - 1

(2) The sum of the digits of x equals 6k

(1) says y=215 Now xy=215*x So it depends on x whether is divisible or not ...> Insufficient
(2) says x is divisible by 3( condition of divisibility by 3 is the sum of the digits shud be divided by 3 and here it is 6k.)

Hence answer shud be B.

Consider KUDOS if its helpful.
Math Expert V
Joined: 02 Sep 2009
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If x, y and k are integers, is xy divisible by 3? (1) y =  [#permalink]

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2
shrive555 wrote:
If x, y and k are integers, is xy divisible by 3?

(1) y = 216 - 1

(2) The sum of the digits of x equals 6k

shrive555 please check the questions when posting. The correct question is as follows:

If x, y and k are integers, is xy divisible by 3?

Now, $$xy$$ will be divisible by 3 if either unknown is divisible by 3.

(1) y = 2^16 - 1 --> $$y =2^{16}-1=(2^8-1)(2^8+1)=(2^4-1)(2^4+1)(2^8+1)=15(2^4+1)(2^8+1)$$, so $$y$$ is a multiple of 3. Sufficient.

(2) The sum of the digits of x equals 6^k --> if $$k\geq{1}$$ then $$x$$ is divisible by 3 (as the sum of the digits of $$x$$ will be multiple of 3) and the answer to the question will be YES but if $$k=0$$ then $$x$$ could be 1 and we won't be sure whether $$x$$ is a multiple of 3. Not sufficient.

Answer: A.
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Re: integers divisibility  [#permalink]

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sorry Bunel: i will take care of that
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Re: integers divisibility  [#permalink]

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Bunuel wrote:
shrive555 wrote:
If x, y and k are integers, is xy divisible by 3?
(2) The sum of the digits of x equals 6^k --> if $$k\geq{1}$$ then $$x$$ is divisible by 3 (as the sum of the digits of $$x$$ will be multiple of 3) and the answer to the question will be YES but if $$k=0$$ then $$x$$ could be 1 and we won't be sure whether $$x$$ is a multiple of 3. Not sufficient.

Answer: A.

I need help understanding the red part.
I thought that if k=0, then the sum of digits of x will be zero (since the sum is a function of k), which implies that x is 0.
Maybe I am not reading the prompt correctly.
Thanks
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Re: integers divisibility  [#permalink]

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@Eden, any real number raised to 0 will be = 1, and 1 is not a multiple of 3.
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GMAT 1: 770 Q50 V47 Re: integers divisibility  [#permalink]

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1
1. We can observe a pattern.
2^1 - 1 = 1 Not divisible
2^2 - 1 =3 Divisible
2^3 - 1 = 7 Not Divisible
2^4 - 1 =15 Divisible
2^6 - 1 = 63

2 rasied to even power minus one is always divisible by 3. Hence y= 2^16-1 is divisible. If y is divisible by 3, so is xy. Hence sufficient

2. If sum of digits of x is equal to 6^k, where k is an integer, k can be 0 or a positive integer. (cant be negative as that will lead to a fractional value for 6^k which cant be possible as the sum of the digits) now if k is a positive integer- 6^k is always going to be divisible by 3, and since sum of digits is divisible by 3, so will be x, and hence so will be xy. But what if k is 0? Then 6^0 becomes 1, and all info (2) gives us is that sum of digits of x is divisible by 1 - so what ? all numbers are divisible by 1. this cant give enough information to determine whether xy is divisible by 3 or not. hence insufficient.
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Re: integers divisibility  [#permalink]

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Eden,

when k>=1, statement 2 is true and divisible by 3.
but when k=0 ,the sum of the digits equals to 6^0 = 1 ( not zero) , which is not divisible by 3. Thats why option 2 is not sufficient as we dont know whether k >=1 or k =0 or k<0.

Eden wrote:
Bunuel wrote:
shrive555 wrote:
If x, y and k are integers, is xy divisible by 3?
(2) The sum of the digits of x equals 6^k --> if $$k\geq{1}$$ then $$x$$ is divisible by 3 (as the sum of the digits of $$x$$ will be multiple of 3) and the answer to the question will be YES but if $$k=0$$ then $$x$$ could be 1 and we won't be sure whether $$x$$ is a multiple of 3. Not sufficient.

Answer: A.

I need help understanding the red part.
I thought that if k=0, then the sum of digits of x will be zero (since the sum is a function of k), which implies that x is 0.
Maybe I am not reading the prompt correctly.
Thanks
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Posts: 12092
Re: If x, y and k are integers, is xy divisible by 3? (1) y =  [#permalink]

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_________________ Re: If x, y and k are integers, is xy divisible by 3? (1) y =   [#permalink] 14 Aug 2018, 03:13
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