Mathematical Communication Math Example 4

Solution

1. 1
   Claim: For any integer $n$, $6 \mid n(n+1)(n+2)$.
2. 2
   Proof: Among any three consecutive integers $n, n+1, n+2$, at least one is even (divisible by 2) since every second integer is even.
3. 3
   Among any three consecutive integers, exactly one is divisible by 3 (since $3 \mid (n+k)$ for exactly one $k \in \{0,1,2\}$).
4. 4
   Therefore $n(n+1)(n+2)$ is divisible by both 2 and 3. Since $\gcd(2,3)=1$, it is divisible by $2 \times 3 = 6$. $\square$