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n and m are positive integers, can n divisible by 3? 1/ n [#permalink ]
24 Sep 2009, 12:58

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n and m are positive integers, can n divisible by 3? 1/ n divisible by m(m^2+2) 2/ n divisible by m^2(m+2) I guess it's C ???

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Re: Divisible question [#permalink ]
24 Sep 2009, 17:18
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The answer is A . Any integer can be represented in terms of divisibility by 3 by placing them in the following 3 groups: a) (3x) - Divisible by 3 (ex. 3, 6, 9, etc.) b) (3x + 1) - Remainder of 1 (ex. 4, 7, 10, etc.) c) (3x + 2) - Remainder of 2 (ex. 5, 8, 11, etc.) where x is an integer. Statement 1: n divisible by m(m^2+2): a) Let m = 3xm(m^2 + 2) = (3x)((3x)^2 + 2) = 3(x)(9x^2 + 2) DIVISIBLE BY 3 b) Let m = 3x + 1m(m^2 + 2) = (3x+1)((3x+1)^2 + 2) = (3x + 1)(9x^2 + 6x + 1 + 2) = (3x + 1)(9x^2 + 6x + 3) = 3(3x^2 + 2x + 1)(3x + 1) DIVISIBLE BY 3 c) Let m = 3x + 2m(m^2 + 2) = (3x+2)((3x+2)^2 + 2) = (3x + 1)(9x^2 + 12x + 4 + 2) = (3x + 1)(9x^2 + 12x + 6) = 3(3x^2 + 4x + 2)(3x + 1) DIVISIBLE BY 3 Therefore Statement 1 is sufficient. If n is divisible by m(m^2 + 2), then it is divisible by 3. Statement 2: n divisible by m^2(m+2): ... c) Let m = 3x + 2 [m]m^2(m + 2) = (3x+2)(3x+2)(3x+2+2) = (3x+2)(3x+2)(3x+4) NOT DIVISIBLE BY 3 Therefore Statement 2 is not sufficient. The answer is A.

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Re: Divisible question [#permalink ]
24 Sep 2009, 19:29

AKProdigy87 wrote:

The answer is A . Any integer can be represented in terms of divisibility by 3 by placing them in the following 3 groups: a) (3x) - Divisible by 3 (ex. 3, 6, 9, etc.) b) (3x + 1) - Remainder of 1 (ex. 4, 7, 10, etc.) c) (3x + 2) - Remainder of 2 (ex. 5, 8, 11, etc.) where x is an integer. Statement 1: n divisible by m(m^2+2): a) Let m = 3xm(m^2 + 2) = (3x)((3x)^2 + 2) = 3(x)(9x^2 + 2) DIVISIBLE BY 3 b) Let m = 3x + 1m(m^2 + 2) = (3x+1)((3x+1)^2 + 2) = (3x + 1)(9x^2 + 6x + 1 + 2) = (3x + 1)(9x^2 + 6x + 3) = 3(3x^2 + 2x + 1)(3x + 1) DIVISIBLE BY 3 c) Let m = 3x + 2m(m^2 + 2) = (3x+2)((3x+2)^2 + 2) = (3x + 1)(9x^2 + 12x + 4 + 2) = (3x + 1)(9x^2 + 12x + 6) = 3(3x^2 + 4x + 2)(3x + 1) DIVISIBLE BY 3 Therefore Statement 1 is sufficient. If n is divisible by m(m^2 + 2), then it is divisible by 3. Statement 2: n divisible by m^2(m+2): ... c) Let m = 3x + 2 [m]m^2(m + 2) = (3x+2)(3x+2)(3x+2+2) = (3x+2)(3x+2)(3x+4) NOT DIVISIBLE BY 3 Therefore Statement 2 is not sufficient. The answer is A.

Hi AKProdigy87,

I don't quite understand that. Say we have n=4 and m=2.

Then m(m^2+2)=2(2^2+2)=12

n=4 is devisible by 12 but doesn't mean 3 is divisible by n.

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Re: Divisible question [#permalink ]
25 Sep 2009, 07:11

hi AKProdigy87 , good way of solving, but is this kind of substitution correct?

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Re: Divisible question [#permalink ]
25 Sep 2009, 10:34

swat wrote:

AKProdigy87 wrote:

The answer is A . Any integer can be represented in terms of divisibility by 3 by placing them in the following 3 groups: a) (3x) - Divisible by 3 (ex. 3, 6, 9, etc.) b) (3x + 1) - Remainder of 1 (ex. 4, 7, 10, etc.) c) (3x + 2) - Remainder of 2 (ex. 5, 8, 11, etc.) where x is an integer. Statement 1: n divisible by m(m^2+2): a) Let m = 3xm(m^2 + 2) = (3x)((3x)^2 + 2) = 3(x)(9x^2 + 2) DIVISIBLE BY 3 b) Let m = 3x + 1m(m^2 + 2) = (3x+1)((3x+1)^2 + 2) = (3x + 1)(9x^2 + 6x + 1 + 2) = (3x + 1)(9x^2 + 6x + 3) = 3(3x^2 + 2x + 1)(3x + 1) DIVISIBLE BY 3 c) Let m = 3x + 2m(m^2 + 2) = (3x+2)((3x+2)^2 + 2) = (3x + 1)(9x^2 + 12x + 4 + 2) = (3x + 1)(9x^2 + 12x + 6) = 3(3x^2 + 4x + 2)(3x + 1) DIVISIBLE BY 3 Therefore Statement 1 is sufficient. If n is divisible by m(m^2 + 2), then it is divisible by 3. Statement 2: n divisible by m^2(m+2): ... c) Let m = 3x + 2 [m]m^2(m + 2) = (3x+2)(3x+2)(3x+2+2) = (3x+2)(3x+2)(3x+4) NOT DIVISIBLE BY 3 Therefore Statement 2 is not sufficient. The answer is A.

Hi AKProdigy87,

I don't quite understand that. Say we have n=4 and m=2.

Then m(m^2+2)=2(2^2+2)=12

n=4 is devisible by 12 but doesn't mean 3 is divisible by n.

The q is saying n is divisible by m(m^2+2) if m is 12 then the minimum value of n is 12 which is divisible by 3.

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Re: Divisible question [#permalink ]
25 Sep 2009, 22:39

Yes. Statement 1 alone is sufficient taking m(m^2+2) and since m is positive integer, you can take m = 1,2,3 and so on for any such value of m, this equation m(m^2+2) will always be a multiple of 3. Since its a multiple of 3 and is a factor of n, thus n will be a multiple of 3 or in other words n will be divisible b 3. Statement 2 is not sufficient taking m^2(m+2) and since m is positive integer, you can take m = 1,2,3 and so on for any such value of m, this equation m^2(m+2), seems to be a multiple of 3 in some cases and not a multiple in some cases. So based on this alone we cannot guess if n is divisible by 3 or not. Hence A.

Re: Divisible question
[#permalink ]
25 Sep 2009, 22:39

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