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Q: (n^3 - n)/4 = int ?
or n(n^2-1)/4 = int?
or (n-1)n(n+1)/4 = int?
Since n is an int, (n-1), n, (n+1) indicate consecutive int's. Which means if n is ODD then it's SUFF

1) says n = 2K+1 that is 3,5,7... when K=1,2,3...
Since it's saying n is ODD, SUFF => AD

2) says (n^2 + n)/6 = int
or n(n+1)/6=int
This can be true when n=5 or n=6 (among other values of course)
So we don't know for sure if n is ODD
NOT SUFF => A

If n is a positive integer, is n^3 - n divisible by 4?

(1) n = 2K + 1, where k is an integer (2) n^2 + n is divisible by 6

(1) n= 2k+1
n^3-n= n(n^2-1)= n (n-1)(n+1) = (2k+1)( 2k)(2k+2)= 4k ( 2k+1) (k+1)
since k is integer ==> n is divisible by 4 ---> suff
(2)n= 2---> n^2+n= 6
n^3-n= 6 is not divisible by 4----> insuff

Btw, since the product of three consecutive numbers is divisible by 6. The information in 2 is useless, it doesn't provide further information to conclude that n^3-n is divisible by 4

Q: (n^3 - n)/4 = int ? or n(n^2-1)/4 = int? or (n-1)n(n+1)/4 = int? Since n is an int, (n-1), n, (n+1) indicate consecutive int's. Which means if n is ODD then it's SUFF

1) says n = 2K+1 that is 3,5,7... when K=1,2,3... Since it's saying n is ODD, SUFF => AD

2) says (n^2 + n)/6 = int or n(n+1)/6=int This can be true when n=5 or n=6 (among other values of course) So we don't know for sure if n is ODD NOT SUFF => A

Great explanation. Could you explain why you say when n is odd, its divisible by 4?

Q: (n^3 - n)/4 = int ? or n(n^2-1)/4 = int? or (n-1)n(n+1)/4 = int? Since n is an int, (n-1), n, (n+1) indicate consecutive int's. Which means if n is ODD then it's SUFF

1) says n = 2K+1 that is 3,5,7... when K=1,2,3... Since it's saying n is ODD, SUFF => AD

2) says (n^2 + n)/6 = int or n(n+1)/6=int This can be true when n=5 or n=6 (among other values of course) So we don't know for sure if n is ODD NOT SUFF => A

Great explanation. Could you explain why you say when n is odd, its divisible by 4?

Q: (n^3 - n)/4 = int ? or n(n^2-1)/4 = int? or (n-1)n(n+1)/4 = int? Since n is an int, (n-1), n, (n+1) indicate consecutive int's. Which means if n is ODD then it's SUFF

1) says n = 2K+1 that is 3,5,7... when K=1,2,3... Since it's saying n is ODD, SUFF => AD

2) says (n^2 + n)/6 = int or n(n+1)/6=int This can be true when n=5 or n=6 (among other values of course) So we don't know for sure if n is ODD NOT SUFF => A

Great explanation. Could you explain why you say when n is odd, its divisible by 4?

Because for any 3 consec positive int's if the middle number is odd then the the other two are even, which means the other two numbers are each divisible by 2.. their product effectively is then divisible by 2*2 or 4.

Q: (n^3 - n)/4 = int ? or n(n^2-1)/4 = int? or (n-1)n(n+1)/4 = int? Since n is an int, (n-1), n, (n+1) indicate consecutive int's. Which means if n is ODD then it's SUFF

1) says n = 2K+1 that is 3,5,7... when K=1,2,3... Since it's saying n is ODD, SUFF => AD

2) says (n^2 + n)/6 = int or n(n+1)/6=int This can be true when n=5 or n=6 (among other values of course) So we don't know for sure if n is ODD NOT SUFF => A

Great explanation. Could you explain why you say when n is odd, its divisible by 4?

your can refer to my above explanation

Actually that's how I solved the problem too. But I wanted to see if there are other ways.

I still wanna know why when n is odd, its divisible by 4.....

If n is a positive integer, is n^3 - n divisible by 4?

(1) n = 2K + 1, where k is an integer (2) n^2 + n is divisible by 6

n^3 -n can be written as n(n^2-1) = n(n+1)(n-1), if n = (2K+1) then
n^3 -n = (2K+1)(2K+2)(2K)
If K = 1 then n^3-n = 3*4*1 divisible by 4
If k = 2 then n^3-n = 5*6*4 divisible by 4
if k =3 then n^3-n = 7*8*6 divisible by 4
if k =4 then n^3 -n is divisible by 4.
So A is sufficient.

From statement 2, we get n(n+1) = 0 mod 6 then if (n^3-n) is divisible by 4 then (n-1) must be divisible by 2. If n is odd then this is true, but if n is even then it is false. So statement 2 is not sufficient.

I don't think there're any difference between " be a multiple of" and " be divisible by"!
"be a multiple of" means that n can be written n= xk ( x is integer)
"be divisible by" means that there's a x ( an integer) that multiples with k yield n.

These are the most basic principles. I suggest people who feel not clear on these principles download the file attached in the opening post of that thread and read through it. _________________

Keep on asking, and it will be given you;
keep on seeking, and you will find;
keep on knocking, and it will be opened to you.

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