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Are i, j, k consecutive integers?

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Are i, j, k consecutive integers?  [#permalink]

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New post 26 Sep 2010, 05:29
4
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A
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D
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Question Stats:

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Are i, j, k consecutive integers?

(1) The remainder when i + j + k is divided by 3 is 2
(2) The remainder when i*j*k is divided by 3 is 1
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Re: Consecutive integers  [#permalink]

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New post 26 Sep 2010, 05:40
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rxs0005 wrote:
Are i , j , k consecutive integers


The remainder when i + j + k is divided by 3 is 2

The remainder when i*j*k is divided by 3 is 1


I think it's meant that \(i\), \(j\), and \(k\) are integers.

NOTES:
Out of any 3 consecutive integers exactly one is divisible by 3. So the product of ANY 3 consecutive integers is divisible by 3 (so remainder is zero).

Consecutive integers can be represented as ... \(n-1\), \(n\), \(n+1\), ... where \(n\) is an integer. Sum of 3 consecutive integers would be \((n-1)+n+(n+1)=3n\), so the sum of ANY 3 consecutive integers is divisible by 3 (so remainder is zero).

More generally:
• If \(k\) is odd, the sum of \(k\) consecutive integers is always divisible by \(k\). Given \(\{9,10,11\}\), we have \(k=3\) consecutive integers. The sum of 9+10+11=30, therefore, is divisible by 3.

• If \(k\) is even, the sum of \(k\) consecutive integers is never divisible by \(k\). Given \(\{9,10,11,12\}\), we have \(k=4\) consecutive integers. The sum of 9+10+11+12=42, therefore, is not divisible by 4.

• The product of \(k\) consecutive integers is always divisible by \(k!\), so by \(k\) too. Given \(k=4\) consecutive integers: \(\{3,4,5,6\}\). The product of 3*4*5*6 is 360, which is divisible by 4!=24.

(1) The remainder when i + j + k is divided by 3 is 2 --> as the remainders is not zero, then \(i\), \(j\), and \(k\) are not consecutive integers. Sufficient.

(2) The remainder when i*j*k is divided by 3 is 1--> as the remainders is not zero, then \(i\), \(j\), and \(k\) are not consecutive integers. Sufficient.

Answer: D.

Hope it helps.
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Re: Are i, j, k consecutive integers?  [#permalink]

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New post 29 Dec 2016, 01:46
Excellent Question.
Here is what i did in this one =>

RULE -> Sum and Product of 3 consecutive integers is always divisible by 3.

Statement 1=>
Sum =3k+2 => They can never be consecutive
Statement 2=>
Product =3k+1
Hence they can never be consecutive.

Hence D

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Re: Are i, j, k consecutive integers?  [#permalink]

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New post 21 Dec 2018, 05:42
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Re: Are i, j, k consecutive integers?   [#permalink] 21 Dec 2018, 05:42
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