Question

If $n$ is an even natural number, then the largest natural number by which $n\left( n+1 \right)\left( n+2 \right)$ is divisible is\[\]
A.6\[\]
B.8\[\]
C.12\[\]
D. 24\[\]

Answer 1

Hint: Put $n=2m$ to find whether the expression is divisible by 4. Use the fact that the product of two consecutive natural numbers is divisible by 2 and product of 3 natural numbers is divisible 3. Finally, use the property that if natural numbers $a$ and $b$ are factors of $c$ then $ab$ is also a factor.\[\]

Complete step-by-step answer:
We know that the natural number $c$is divisible by a non-zero natural number s $a$ and $b$ if and only if $c$is divisible by $a\times b=ab$.\[\]
An even natural number is a natural number is exactly divisible by 2 in other words a multiple of 2. So if any natural number says $n$ is even natural number the we can express $2|m\Rightarrow n=2m$ for natural number $m$.
The given expression is (denoted as ${{P}_{n}}$, $n\in N$ )
\[{{P}_{n}}=n\left( n+1 \right)\left( n+2 \right)\]
Let us substitute $n = 2m$ in the above expression and get ,
\[\begin{align}
  & {{P}_{n}}=2m\left( 2m+1 \right)\left( 2m+2 \right) \\
 & \Rightarrow {{P}_{n}}=2m\left( 2m+1 \right)2\left( m+1 \right) \\
 & \Rightarrow {{P}_{n}}=4m\left( m+1 \right)\left( 2m+1 \right)={{P}_{m}}\left( say \right) \\
\end{align}\]

So ${{P}_{m}}$ is divisible by 4 then ${{P}_{n}}$ is also divisible 4. We also see in the expression of ${{P}_{m}}$ that the two terms multiplied with each other $m\left( m+1 \right)$ which is an expression for any two consecutive natural numbers. We know that if we pick any consecutive two natural ones one of them must be a multiple of 2. So the expression ${{P}_{m}}$ is divisible by 2 . So ${{P}_{m}}$ is divisible by $2\times 4=8$.\[\]

We know that if we pick any three consecutive natural numbers then one of them must be multiple of 3 .We can see that given expression ${{P}_{n}}=n\left( n+1 \right)\left( n+2 \right)$ is expression for the product of any 3 consecutive natural numbers which we now know to be divisible by 3. So ${{P}_{n}}$ is divisible by 3. So ${{P}_{n}}={{P}_{m}}$ is divisible by $8\times 3=24$.\[\]
So the largest natural number that exactly divides $n\left( n+1 \right)\left( n+2 \right)$ is 24 and the correct option is D. \[\]

So, the correct answer is “Option D”.

Note: It is to be noted that unlike product, sum does not follow the property of divisibility to individual numbers. In symbols, $a+b|c$then it is not necessarily true that$a|c,b|c$. We can also prove the result $24|n\left( n+1 \right)\left( n+2 \right)$with induction where we have to prove ${{P}_{1}}$, assume ${{P}_{k}}$ and then prove ${{P}_{k}}\Rightarrow {{P}_{k+1}}$ .