Question

Prove that one of
of every three consecutive integer
is divisible by
3.​

Answer 1

Let three consecutive positive integers be

1) n

2) n + 1

3) n + 2.

Whenever a number is divided by 3, the remainder obtained is either 0 or 1 or 2.

∴ n = 3p or 3p + 1 or 3p + 2, where p is some integer.

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

Answer 2

Answer:

Let n,n+1,n+2 be three consecutive positive integers.

We know that n is of the form 3q,3q+1 or, 3q+2

So, we have the following

Case I When n=3q

In this case, n is divisible by 3 but n+1 and n+2 are not divisible by 3.

Case II When n=3q+1

In this case, n+2=3q+1+2=3 is divisible by 3 but n and n+1 are not divisible by 3.

Case III When n=3q+2

In this case, n+1=3q+1+2=3(q+1) is divisible by 3 but n and n+2 are not divisible by 3.

Hence one of n,n+1 and n+2 is divisible by 3.

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