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# Are i, j, k consecutive integers?

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Director
Joined: 07 Jun 2004
Posts: 599
Location: PA
Are i, j, k consecutive integers?  [#permalink]

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26 Sep 2010, 04:29
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Difficulty:

35% (medium)

Question Stats:

73% (01:02) correct 27% (01:20) wrong based on 255 sessions

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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
Math Expert
Joined: 02 Sep 2009
Posts: 52344

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26 Sep 2010, 04:40
2
7
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.

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

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29 Dec 2016, 00:46
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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21 Dec 2018, 04:42
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Re: Are i, j, k consecutive integers? &nbs [#permalink] 21 Dec 2018, 04:42
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