Question

If n is an even natural number, then the largest natural number by which n(n + 1)(n + 2) is divisible, is​

Answer 1

Answer: Out of n and n+2, one is divisible by 2 and the other by 4, hence n(n+2) is divisible by 8.

Also n,n+1,n+2 are three consecutive numbers, hence one of them is divisible by 3.

Hence, n(n+1)(n+2) must be divisible by 24. This will be true for any even number n

Answer 2

Answer:

The largest natural number that divides n(n + 1)(n + 2) is 24.

Step-by-step explanation:

Given:-

n is an even natural number.

To find:-

The largest natural number which divides the number n(n + 1)(n + 2).

According to the question,

Since n is an even number.

So, the number n is divisible by $2$.

⇒ (n + 2) is divisible by 4.

Or,

If n + 2 is divisible by 2, then the number n is divisible by 4.

This means n(n + 2) is divisible by the product of 4 and 2, i.e.,

n(n + 2) is divisible by 8.

Notice that n, n + 1 and n + 2 are three consecutive numbers.

Thus, $1$ of them must be divisible by $3$.

This implies n(n + 1)(n + 2) is divisible by the product of 8 and 3, i.e.,

n(n + 1)(n + 2) is divisible by $24$.

Therefore, the largest natural number that divides n(n + 1)(n + 2) is 24.

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