If n is an even natural number, then the largest natural number by which n(n+1)(n+2) is divisible by 24. Choose whether the given statement is true or false.

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True

False

answer is A.

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Detailed Solution

An even natural number is a natural number that is exactly divisible by 2, in other words a multiple of 2. So if any natural number, say n, is even natural number the we can express 2|m⇒n=2m for natural number m.
The given expression is (denoted as Pn, n∈N )
Pn=n(n+1)(n+2)
Let us substitute n=2m in the above expression and get ,
Pn=2m(2m+1)(2m+2)
⇒Pn=2m(2m+1)2(m+1)
⇒Pn=4m(m+1)(2m+1)=Pm(say)
So Pm is divisible by 4 then Pn is also divisible 4. We also see in the expression of Pm that the two terms multiplied with each other m(m+1) 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 Pm is divisible by 2 . So Pm is divisible by 2×4=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 Pn=n(n+1)(n+2) is expression for the product of any 3 consecutive natural numbers which we now know to be divisible by 3. So Pn is divisible by 3. So Pn=Pm is divisible by 8×3=24.
So the largest natural number that exactly divides n(n+1)(n+2) is 24.
Hence, the given statement is true.

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