Divisible by Six

• Book: A Walk through Combinatorics
• Induction

Problem

Prove that for all natural numbers $n$, the number $a(n) = n^3 + 11n$ is divisible by $6$.

Proof

By Induction

For $n = 1$, $a_n = 12$, which is divisible by $6$.

Now the induction step, suppose $a_n$ is divisible by $6$.

Each of these is divisible by 6.

• $a_n$ based on the induction hypothesis
• $12$ is trivial
• $3n(n+1)$, either $n$ or $n+1$ is odd and divisible by 2, hence this can be written as $6x$o

Hence proved