Question

Show the product of three consecutive natural number which is divisible by 6

Answer 1

Answer:

Step-by-step explanation:

Let us three consecutive  integers be, n, n + 1 and n + 2.

Whenever a number is divided by 3

the remainder obtained is either 0 or 1 or 2.    

let n = 3p or 3p + 1 or 3p + 2, where p is some integer.(Euclid's division Lemma)

If n = 3p, then n is divisible by 3.

If n = 3p + 1, then n + 2 = 3p + 1 + 2 = 3p + 3 = 3(p + 1) is divisible by 3.

If n = 3p + 2, then n + 1 = 3p + 2 + 1 = 3p + 3 = 3(p + 1) is divisible by 3.

So that n, n + 1 and n + 2 is always divisible by 3.

⇒ n (n + 1) (n + 2) is divisible by 3. 

Similarly, whenever a number is divided 2 we will get the remainder is 0 or 1.

∴ n = 2q or 2q + 1, where q is some integer.

If n = 2q, then n and n + 2 = 2q + 2 = 2(q + 1) are divisible by 2.

If n = 2q + 1, then n + 1 = 2q + 1 + 1 = 2q + 2 = 2 (q + 1) is divisible by 2.

So that n, n + 1 and n + 2 is always divisible by 2.

⇒ n (n + 1) (n + 2) is divisible by 2. 

But n (n + 1) (n + 2) is divisible by 2 and 3. 

∴ n (n + 1) (n + 2) is divisible by 6.

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