Hi maxim 730,
let's try to guess if positive integer n – 1 a multiple of 3?
REGRADING THE QUESTIONS FIRST ASSUMPTION :
(1) n3 – n is a multiple of 3 : this means that n(n-1)(n+1) IS MULTIPLE OF 3 : this means that each one could suffisantly be a multiple of three. So the assumption 1 is insuffisant ( so answer is either B, D or E).
To make sure of this first answer, you can pick n = 2. or n=3; both possibilities give you a number of ( n^3 - n) that is multiple of 3;
REGARDING THE ASSUMPTION 2: (2) says that n3 + 2n2+ n is a multiple of 3
this leads to n(n+1)^2 is multiple of 3 , we, here, have two possibilities, either n is multiple of 3 or (n+1)^2 is multiple of 3
let's take look at each of these conclusions :
* if n is multiple of 3, then n-1 is not multiple of 3 ( to rephrase, we would say that the remainder of the division of n-1 by 3 is 2)
* If (n+1)^2 is multiple of 3, ( to rephrase, 3 is prime factor of (n+1)2) , then necessarely, n+1 is multilpe 3, because if it wasn't not a prime factor of (n+1), it would'nt be a factor of (n+1)^2, ALL THAT means that (n+1) is multiple of 3, and consequently ( n-1) is not a multiple of 3.
So I would go for B;
I appplogize for my wordy and awkward sentences, you would understand that english my third langage.
PLEASE CAN ANYBODY COMMENT MY REASONING,
good luck for us.
