Statement 1: n^3 - n is a multiple of 3.garuhape wrote:Hi there,
Is positive integer n - 1 a multiple of 3?
(1) n^3 - n is a multiple of 3
(2) n^3 + 2n^2+ n is a multiple of 3
n^3 - n
= n(n^2 - 1)
= n(n+1)(n-1)
The above expression represents 3 consecutive integers: n-1, n, n+1.
Among every 3 consecutive integers, exactly 1 will be a multiple of 3.
Thus, n-1 could be a multiple of 3.
If n is a multiple of 3, then n-1 is not a multiple of 3.
Insufficient.
Statement 2: n^3 + 2n^2+ n is a multiple of 3.
n^3 + 2n^2 + n
= n(n^2 + 2n + 1)
= n(n+1)(n+1)
The above expression represents 2 consecutive integers: n and n+1.
Either n or n+1 must be a multiple of 3.
If n is a multiple of 3, then n-1 is not a multiple of 3.
If n+1 is a multiple of 3, then n-1 is not a multiple of 3.
Since in each case n-1 is not a multiple of 3, sufficient.
The correct answer is B.
