Divisible or not?

Number Theory Level 3

Find the set of all integers $n$ such that the number $n^3 - 9n + 27$ is divisible by 81.

Null set

(100, 400) - (130, 150)

All negative integers All positive integers

1 solution

Kushal Bose
Aug 3, 2016

Let, $n^3-9 n +27=81 p$

$n^3 -9 n=81 p + 54$

$n^3=9 (n+ 81 p + 6)$

So, $9 | n^3$

if $n=3 k$ it will be divisible by $27$ but not divisible by $81$

because $27 k^3 -27 k +27 = 27(k(k-1)(k+1) +1)$

The first part is divisible by $6$

So, there is no value of $n$

I think there is an error while going from step 1 to 2.

Rushikesh Jogdand - 4 years, 2 months ago